Let be an order of an imaginary quadratic field. The class equation of is the equation , where is the extension field minimal polynomial of over , with the -invariant of . (If has generator , then . The degree of is equal to the class number of the field of fractions of .The polynomial is also called the class equation of (e.g., Cox 1997, p. 293).It is also true thatwhere the product is over representatives of each ideal class of .If has discriminant , then the notation is used. If is not divisible by 3, the constant term of is a perfect cube. The table below lists the first few class equations as well as the corresponding values of , with being generators of ideals in each ideal class of . In each case, the constant term is written out as a cube times a cubefree part.0..
A Proth number that is prime, i.e., a number of the form for odd , a positive integer, and . Factors of Fermat numbers are of this form as long as they satisfy the condition odd and . For example, the factor of is not a Proth prime since . (Otherwise, every odd prime would be a Proth prime.)Proth primes satisfy Proth's theorem, i.e., a number of this form is prime iff there exists a number a such that is congruent to modulo . This provides an easy computational test for Proth primes. Yves Gallot has written a downloadable program for testing Proth primes and many of the largest currently known primes have been found with this program.A Sierpiński number of the second kind is a number satisfying Sierpiński's composite number theorem, i.e., a Proth number such that is composite for every .The first few Proth primes are 3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, ...(OEIS A080076).The following table gives the first few indices such that is prime for..
A colossally abundant number is a positive integer for which there is a positive exponent such thatfor all . All colossally abundant numbers are superabundant numbers.The first few are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 160626866400, ... (OEIS A004490). The following table lists the colossally abundant numbers up to , as given by Alaoglu and Erdős (1944).factorization of 221.50062.000122.333602.8001203.0003603.25025203.71450403.838554404.1877207204.50914414404.58143243204.699216216004.8553675672005.14169837768005.4121606268664005.6473212537328005.69293163582512005.8882888071057872006.07820216497405104006.18760649492215312006.2382244031211966544006.407The first 15 elements of this sequence agree with those of the superiorhighly composite numbers (OEIS A002201).The th colossally abundant number has the form , where..
Let be the sum of the infinitary divisors of a number . An infinitary perfect number is a number such that . The first few are 6, 60, 90, 36720, ... (OEIS A007357). Cohen (1990) found 14 such numbers, and 155 are known as of January 2004 (Pedersen).
A concave polygon is a polygon that is not convex.A simple polygon is concave iff at least one of its internal angles is greater than . An example of a non-simple (self-intersecting) polygon is a star polygon.A concave polygon must have at least four sides.
A qubit (or quantum bit) is the analog of a bit for quantum computation. Unlike an ordinary bit, which may only assume two possible values (usually called 0 and 1), a qubit may assume a continuum of values of the form where and are arbitrary complex numbers satisfying .
The qubit can be represented as a point on a unit sphere called the Bloch sphere. Define the angles and by letting and . Here, is taken to be real, which can always be made real by multiplying by an overall phase factor (that is unobservable). Then is represented by the unit vector (, , ) called the Bloch vector.
Let and be any functions of a set of variables . Then the expression(1)is called a Poisson bracket (Poisson 1809; Whittaker 1944, p. 299). Plummer (1960, p. 136) uses the alternate notation .The Poisson brackets are anticommutative,(2)(Plummer 1960, p. 136).Let be independent functions of the variables . Then the Poisson bracket is connected with the Lagrange bracket by(3)where is the Kronecker delta. But this is precisely the condition that the determinants formed from them are reciprocal (Whittaker 1944, p. 300; Plummer 1960, p. 137).If and are physically measurable quantities (observables) such as position, momentum, angular momentum, or energy, then they are represented as non-commuting quantum mechanical operators in accordance with Heisenberg's formulation of quantum mechanics. In this case,(4)where is the commutator and is the Poisson bracket. Thus, for example, for a single particle..
If a discrete group of displacements in the plane has more than one center of rotation, then the only rotations that can occur are by 2, 3, 4, and 6. This can be shown as follows. It must be true that the sum of the interior angles divided by the number of sides is a divisor of .where is an integer. Therefore, symmetry will be possible only forwhere is an integer. This will hold for 1-, 2-, 3-, 4-, and 6-fold symmetry. That it does not hold for is seen by noting that corresponds to . The case requires that (impossible), and the case requires that (also impossible).The point groups that satisfy the crystallographic restriction are called crystallographic point groups.Although -fold rotations for differing from 2, 3, 4, and 6 are forbidden in the strict sense of perfect crystallographic symmetry, there are exotic materials called quasicrystals that display these symmetries. In 1984, D. Shechtman discovered a class of aluminum alloys whose X-ray..
Let be a group with normal series (, , ..., ). A normal factor of is a quotient group for some index . is a solvable group iff all normal factors are Abelian.
An inner automorphism of a group is an automorphism of the form , where is a fixed element of .The automorphism of the symmetric group that maps the permutation to is an inner automorphism, since .
A number given by the generating function(1)It satisfies , , and even coefficients are given by(2)(3)where is a Bernoulli number and is an Euler polynomial.The first few Genocchi numbers for , 4, ... are , 1, , 17, , 2073, ... (OEIS A001469).The first few prime Genocchi numbers are and 17, which occur for and 8. There are no others with (Weisstein, Mar. 6, 2004). D. Terr (pers. comm., Jun. 8, 2004) proved that these are in fact, the only prime Genocchi numbers.
The limiting rabbit sequence written as a binary fraction (OEIS A005614), where denotes a binary number (a number in base-2). The decimal value is(1)(OEIS A014565).Amazingly, the rabbit constant is also given by the continued fraction [0; , , , , ...] = [2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184, ...] (OEIS A000301), where are Fibonacci numbers with taken as 0 (Gardner 1989, Schroeder 1991). Another amazing connection was discovered by S. Plouffe. Define the Beatty sequence by(2)where is the floor function and is the golden ratio. The first few terms are 1, 3, 4, 6, 8, 9, 11, ... (OEIS A000201). Then(3)This is a special case of the Devil's staircase function with .The irrationality measure of is (D. Terr, pers. comm., May 21, 2004).
A complex rotation is a map of the form , where is a real number, which corresponds to counterclockwise rotation by radians about the origin of points the complex plane.
A complex magnification is a map of the form , where is a positive real number, which corresponds to magnification about the origin of points in the complex plane by the factor if is greater than 1, or shrinking by a factor if is less than 1.
A linear code is cyclic if for every codeword in , the codeword is also in .
A codeword is an element of an error-correcting code . If has length , then a codeword in has the form , where each is a letter in the alphabet of .
Let be an error-correcting code consisting of codewords,in which each codeword consists of letters taken from an alphabet of length , and every two distinct codewords differ in at least places. Then is said to be perfect if for every possible word of length with letters in , there is a unique code word in in which at most letters of differ from the corresponding letters of .It is straightforward to show that is perfect ifIf is a binary linear code, then and , where is the number of generators of , in which case is perfect ifHamming codes and the Golaycode are the only nontrivial examples of perfect codes.
Given a linear code of length and dimension over the field , a parity check matrix of is a matrix whose rows generate the orthogonal complement of , i.e., an element of is a codeword of iff . The rows of generate the null space of the generator matrix .
The Golay code is a perfect linear error-correcting code. There are two essentially distinct versions of the Golay code: a binary version and a ternary version.The binary version is a binary linear code consisting of codewords of length 23 and minimum distance 7. The ternary version is a ternary linear code, consisting of codewords of length 11 with minimum distance 5.A parity check matrix for the binary Golay code is given by the matrix , where is the identity matrix and is the matrixBy adding a parity check bit to each codeword in , the extended Golay code , which is a nearly perfect binary linear code, is obtained. The automorphism group of is the Mathieu group .A second generator is the adjacency matrix for the icosahedron, with appended, where is a unit matrix and is an identity matrix.A third generator begins a list with the 24-bit 0 word (000...000) and repeatedly appends first 24-bit word that has eight or more differences from all words in the list...
A superior highly composite number is a positive integer for which there is an such thatfor all , where the function counts the divisors of (Ramanujan 1962, pp. 87 and 115). It can be shown that all superior highly composite numbers are highly composite and that the th superior highly composite number has the form , where the factors are prime.The first few superior highly composite numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, ... (OEIS A002201), and the corresponding sequence of primes is 2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, 7, 29, 3, 31, 2, 37, 41, 43, ... (OEIS A000705).
Given an integer , the Payam number is the smallest positive odd integer such that for every positive integer , the number is not divisible by any primes such that the multiplicative order of 2 is less than or equal to . Payam numbers are good candidates for searching for Proth primes, i.e., primes of the form , as well as primes of the form .The first few values of for , 3, ... are 3, 9, 15, 105, 105, 105, 105, 105, 165, 165, 75075, ... (OEIS A083556), and the first few values of are 3, 3, 45, 45, 45, 45, 45, 45, 45, 2145, ... (OEIS A083391).
The silver ratio is the quantity defined by the continuedfraction(1)(2)(Wall 1948, p. 24). It follows that(3)so(4)(OEIS A014176).The sequence , of power fractional parts, where is the fractional part, is equidistributed for almost all real numbers , with the silver ratio being one exception.The more general expressions(5)are sometimes known in general as silver means (Knott). The first few values are summarized in the table below.OEISvalue1A0016221.618033988...2A0141762.414213562...3A0983163.302775637...4A0983174.236067977...5A0983185.192582403...
The Cantor diagonal method, also called the Cantor diagonal argument or Cantor's diagonal slash, is a clever technique used by Georg Cantor to show that the integers and reals cannot be put into a one-to-one correspondence (i.e., the uncountably infinite set of real numbers is "larger" than the countably infinite set of integers). However, Cantor's diagonal method is completely general and applies to any set as described below.Given any set , consider the power set consisting of all subsets of . Cantor's diagonal method can be used to show that is larger than , i.e., there exists an injection but no bijection from to . Finding an injection is trivial, as can be seen by considering the function from to which maps an element of to the singleton set . Suppose there exists a bijection from to and consider the subset of consisting of the elements of such that does not contain . Since is a bijection, there must exist an element of such that . But by the..
There are two different definitions of generalized Fermat numbers, one of which is more general than the other. Ribenboim (1996, pp. 89 and 359-360) defines a generalized Fermat number as a number of the form with , while Riesel (1994) further generalizes, defining it to be a number of the form . Both definitions generalize the usual Fermat numbers . The following table gives the first few generalized Fermat numbers for various bases .OEISgeneralized Fermat numbers in base 2A0002153, 5, 17, 257, 65537, 4294967297, ...3A0599194, 10, 82, 6562, 43046722, ...4A0002155, 17, 257, 65537, 4294967297, 18446744073709551617, ...5A0783036, 26, 626, 390626, 152587890626, ...6A0783047, 37, 1297, 1679617, 2821109907457, ...Generalized Fermat numbers can be prime only for even . More specifically, an odd prime is a generalized Fermat prime iff there exists an integer with and (Broadhurst 2006).Many of the largest known prime numbers are generalized..
A congruent number can be defined as an integer that is equal to the area of a rational right triangle (Koblitz 1993).Numbers such that(1)are also known as congruent numbers. They are a generalization of the congruum problem, which is the case .For example, , the smallest congruent numbers are(2)(3)(4)(5)
A noble number is defined as an irrational number having a continued fraction that becomes an infinite sequence of 1s at some point, The prototype is the inverse of the golden ratio , whose continued fraction is composed entirely of 1s (except for the term), .Any noble number can be written aswhere and are the numerator and denominator of the th convergent of .The noble numbers are a subset of but not a subfield, since there is no subfield lying properly between and . To see this, consider , which must be contained in the same field as but is not a noble number since its continued fraction is .
Every irrational number can be expanded in a unique continued fraction expansionsuch that , , , and for . This continued fraction expansion is known as the nearest integer continued fraction expansion of .For example, the nearest integer continued fraction expansion of eis given by
Computational number theory is the branch of number theory concerned with finding and implementing efficient computer algorithms for solving various problems in number theory. Much progress has been made in this field in recent years, both in terms of improved computer speed and in terms of finding more efficient algorithms. Two important applications of computational number theory are primality testing and prime factorization of large integers.Primality testing is considered easy in the sense that very large general numbers (currently up to 4000 digits or so) can be tested reliably for primality. In fact, on August 6, 2002, Agrawal, Saxena, and Kayal found a polynomial time algorithm for testing and proving the primality of general numbers. Although this algorithm is still impractical, it was a landmark discovery, since polynomial time algorithms are considered easy. On the other hand, factoring is considered hard in the sense that..
Analytic number theory is the branch of number theory which uses real and complex analysis to investigate various properties of integers and prime numbers. Examples of topics falling under analytic number theory include Dirichlet L-series, the Riemann zeta function , the totient function , and the prime number theorem.
A Pisot number is a positive algebraic integer greater than 1 all of whose conjugate elements have absolute value less than 1. A real quadratic algebraic integer greater than 1 and of degree 2 or 3 is a Pisot number if its norm is equal to . The golden ratio (denoted when considered as a Pisot number) is an example of a Pisot number since it has degree two and norm .The smallest Pisot number is given by the positive root (OEIS A060006) of(1)known as the plastic constant. This number was identified as the smallest known by Salem (1944), and proved to be the smallest possible by Siegel (1944).Pisot constants give rise to almost integers. For example, the larger the power to which is taken, the closer , where is the floor function, is to either 0 or 1 (Trott 2004). For example, the spectacular example is within of an integer (Trott 2004, pp. 8-9).The powers of for which this quantity is closer to 0 are 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 17, ... (OEIS A051016), while..
Algebraic number theory is the branch of number theory that deals with algebraic numbers. Historically, algebraic number theory developed as a set of tools for solving problems in elementary number theory, namely Diophantine equations (i.e., equations whose solutions are integers or rational numbers). Using algebraic number theory, some of these equations can be solved by "lifting" from the field of rational numbers to an algebraic extension of .More recently, algebraic number theory has developed into the abstract study of algebraic numbers and number fields themselves, as well as their properties.
Let be a number field and let be an order in . Then the set of equivalence classes of invertible fractional ideals of forms a multiplicative Abelian group called the Picard group of .If is a maximal order, i.e., the ring of integers of , then every fractional ideal of is invertible and the Picard group of is the class group of . The order of the Picard group of is sometimes called the class number of . If is maximal, then the order of the Picard group is equal to the class number of .
A particle is said to be undergoing uniform circular motion if its radius vector in appropriate coordinates has the form , where(1)(2)Geometrically, uniform circular motions means that moves in a circle in the -plane with some radius at constant speed. The quantity is called the angular velocity of . The speed of is(3)and the acceleration of P has constant magnitude(4)and is directed toward the center of the circle traced by . This is called centripetal acceleration.Ignoring the ellipticity of their orbits, planet show nearly uniform circular motion about the Sun. (Although due to orbital inclinations, the orbital planes of the different planets are not necessarily coplanar.)
A representation of a group is faithful if it is one-to-one, i.e., if implies for . Equivalently, is faithful if implies , where is the dimension of , is the identity matrix, and is the identity element of .
A power floor prime sequence is a sequence of prime numbers , where is the floor function and is real number. It is unknown if, though extremely unlikely that, infinite sequences of this type exist. An example having eight consecutive primes is , which gives 2, 5, 13, 31, 73, 173, 409, and 967 and has the smallest possible numerator and denominators for an 8-term sequence (D. Terr, pers. comm., Sep. 1, 2004). D. Terr (pers. comm., Jan. 21, 2003) has found a sequence of length 100.
If is an arbitrary integer relatively prime to and is a primitive root of , then there exists among the numbers 0, 1, 2, ..., , where is the totient function, exactly one number such thatThe number is then called the discrete logarithm of with respect to the base modulo and is denotedThe term "discrete logarithm" is most commonly used in cryptography, although the term "generalized multiplicative order" is sometimes used as well (Schneier 1996, p. 501). In number theory, the term "index" is generally used instead (Gauss 1801; Nagell 1951, p. 112).For example, the number 7 is a positive primitive root of (in fact, the set of primitive roots of 41 is given by 6, 7, 11, 12, 13, 15, 17, 19, 22, 24, 26, 28, 29, 30, 34, 35), and since , the number 15 has multiplicative order 3 with respect to base 7 (modulo 41) (Nagell 1951, p. 112). The generalized multiplicative order is implemented in the Wolfram Language..
Let be a number field, then each fractional ideal of belongs to an equivalence class consisting of all fractional ideals satisfying for some nonzero element of . The number of equivalence classes of fractional ideals of is a finite number, known as the class number of . Multiplication of equivalence classes of fractional ideals is defined in the obvious way, i.e., by letting . It is easy to show that with this definition, the set of equivalence classes of fractional ideals form an Abelian multiplicative group, known as the class group of .
The trivial group, denoted or , sometimes also called the identity group, is the unique (up to isomorphism) group containing exactly one element , the identity element. Examples include the zero group (which is the singleton set with respect to the trivial group structure defined by the addition ), the multiplicative group (where ), the point group , and the integers modulo 1 under addition. When viewed as a permutation group on letters, the trivial group consists of the single element which fixes each letter.The trivial group is (trivially) Abelian and cyclic.The multiplication table for is given below. 111The trivial group has the single conjugacy class and the single subgroup .
In 1979, Conway and Norton discovered an unexpected intimate connection between the monster group and the j-function. The Fourier expansion of is given by(1)(OEIS A000521), where and is the half-period ratio, and the dimensions of the first few irreducible representations of are 1, 196883, 21296876, 842609326, ... (OEIS A001379).In November 1978, J. McKay noticed that the -coefficient 196884 is exactly one more than the smallest dimension of nontrivial representations of the (Conway and Norton 1979). In fact, it turns out that the Fourier coefficients of can be expressed as linear combinations of these dimensions with small coefficients as follows:(2)(3)(4)(5)Borcherds (1992) later proved this relationship, which became known as monstrous moonshine. Amazingly, there turn out to be yet more deep connections between the monster group and the j-function...
Given a ring with identity, the special linear group is the group of matrices with elements in and determinant 1.The special linear group , where is a prime power, the set of matrices with determinant and entries in the finite field . is the corresponding set of complex matrices having determinant . is a subgroup of the general linear group and is a Lie-type group. Both and are genuine Lie groups.
Given a ring with identity, the general linear group is the group of invertible matrices with elements in .The general linear group is the set of matrices with entries in the field which have nonzero determinant.
Let be an extension field of , denoted , and let be the set of automorphisms of , that is, the set of automorphisms of such that for every , so that is fixed. Then is a group of transformations of , called the Galois group of . The Galois group of is denoted or .Let be a rational polynomial of degree and let be the splitting field of over , i.e., the smallest subfield of containing all the roots of . Then each element of the Galois group permutes the roots of in a unique way. Thus can be identified with a subgroup of the symmetric group , the group of permutations of the roots of . If is irreducible, then is a transitive subgroup of , i.e., given two roots and of , there exists an element of such that .The roots of are solvable by radicals iff is a solvable group. Since all subgroups of with are solvable, the roots of all polynomials of degree up to 4 are solvable by radicals. However, polynomials of degree 5 or greater are generally not solvable by radicals since (and the alternating..
A permutation group is a finite group whose elements are permutations of a given set and whose group operation is composition of permutations in . Permutation groups have orders dividing .Two permutations form a group only if one is the identity element and the other is a permutation involution, i.e., a permutation which is its own inverse (Skiena 1990, p. 20). Every permutation group with more than two elements can be written as a product of transpositions.Permutation groups are represented in the Wolfram Language as a set of permutation cycles with PermutationGroup. A set of permutations may be tested to see if it forms a permutation group using PermutationGroupQ[l] in the Wolfram Language package Combinatorica` .Conjugacy classes of elements which are interchangedin a permutation group are called permutation cycles.Examples of permutation groups include the symmetric group (of order ), the alternating group (of order for ),..
A bivariate polynomial is a polynomial in two variables.Bivariate polynomials have the formA homogeneous bivariate polynomial, also called a binaryform, has the formwhere is the degree of . The binary form above is often represents as .
Let be a number field with ring of integers and let be a nontrivial ideal of . Then the ideal class of , denoted , is the set of fractional ideals such that there exists a nonzero element of such that .
An integer is a fundamental discriminant if it is not equal to 1, not divisible by any square of any odd prime, and satisfies or . The function FundamentalDiscriminantQ[d] in the Wolfram Language version 5.2 add-on package NumberTheory`NumberTheoryFunctions` tests if an integer is a fundamental discriminant.It can be implemented as: FundamentalDiscriminantQ[n_Integer] := n != 1&& (Mod[n, 4] == 1 \[Or] ! Unequal[Mod[n, 16], 8, 12])&& SquareFreeQ[n/2^IntegerExponent[n, 2]]The first few positive fundamental discriminants are 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, ... (OEIS A003658). Similarly, the first few negative fundamental discriminants are , , , , , , , , , , , ... (OEIS A003657).
Let be a number ring of degree with imaginary embeddings. Then every ideal class of contains an ideal such thatwhere denotes the norm of .
The ring of integers of a number field , denoted , is the set of algebraic integers in , which is a ring of dimension over , where is the extension degree of over . is also sometimes called the maximal order of .
The ring of integers is the set of integers ..., , , 0, 1, 2, ..., which form a ring. This ring is commonly denoted (doublestruck Z), or sometimes (doublestruck I).More generally, let be a number field. Then the ring of integers of , denoted , is the set of algebraic integers in , which is a ring of dimension over , where is the extension degree of over . is also sometimes called the maximal order of .The Gaussian integers is the ring of integers of , and the Eisenstein integers is the ring of integers of , where is a primitive cube root of unity.
An impossible figure in which a stairway in the shape of a square appears to circulate indefinitely while still possessing normal steps (Penrose and Penrose 1958). The Dutch artist M. C. Escher included a Penrose stairway in his mind-bending illustration "Ascending and Descending" (Bool et al. 1982, p. 321; Forty 2003, Plate 68). Distorted variations of the stairway are also depicted in Escher's "House of Stairs" (Bool et al. 1982, p. 301; Forty 2003, Plate 40).In the 1998 film The Avengers, Uma Thurman is shown walking down a Penrose stairway and ending up back where she began.
Let be an error-correcting code consisting of codewords in which each codeword consists of letters taken from an alphabet of length , and every two distinct codewords differ in at least places. Then is said to be nearly perfect if, for every possible word of length with letters in , there is a codeword in in which at most letters of differ from the corresponding letters of . The codeword is unique if it differs from in fewer than places and there is at most one other codeword differing from in places if differs from in places.A nearly perfect code can be derived from a perfect code by adding a parity check digit to the end of each codeword in , so if is a -perfect binary linear code, then is a -nearly perfect binary linear code. In this way, the nearly perfect extended Golay code can be obtained from the perfect Golay code and the nearly perfect extended Hamming codes from the perfect Hamming codes...
Given a linear code , a generator matrix of is a matrix whose rows generate all the elements of , i.e., if , then every codeword of can be represented asin a unique way, where .An example of a generator matrix is the Golay code, which consists of all possible binary sums of the 11 rows.
A Barker code is a string of digits of length such thatfor all . Barker codes are used for pulse compression of radar signals. There are Barker codes of lengths 2, 3, 4, 5, 7, 11, and 13, and it is conjectured that no longer Barker codes exist. A list of known Barker codes up to reversal of digits and negation is given below.lengthcode2, 34, 571113The number of candidate codes of length is therefore equal to the number of -bead black-white reversible strings 1, 2, 3, 6, 10, 20, 36, 72, ... (OEIS A005418), while the numbers of Barker codes of order , 3, ... are 2, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, and 0 for all higher (OEIS A091704).
For a given monic quarticequation(1)the resolvent cubic is the monic cubic polynomial(2)where the coefficients are given in terms of the by(3)(4)(5)The roots , , and of are given in terms of the roots , , , and of by(6)(7)(8)The resolvent cubic of a quartic equation can be used to solve for the roots of the quartic in terms of the roots of the cubic, which can in turn be solved for using the cubic equation.
The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is denoted and is plotted above (using two different scales) along the real axis. Min Max Re Im In general, is defined over the complex plane for one complex variable, which is conventionally denoted (instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). is implemented in the Wolfram Language as Zeta[s].The plot above shows the "ridges" of for and . The fact that the ridges appear to decrease monotonically for is not..
An inner automorphism of a group is an automorphism of the form , where is a fixed element of . An outer automorphism of is an automorphism which cannot be expressed in this form for , but can be so expressed if belongs to a larger group containing .For example, the automorphism of the symmetric group which maps the permutation to is an inner automorphism, since . However, it is an outer automorphism of the alternating group since does not belong to and there is no element of such that .
Let be a group and be a topological G-set. Then a closed subset of is called a fundamental domain of in if is the union of conjugates of , i.e.,and the intersection of any two conjugates has no interior.For example, a fundamental domain of the group of rotations by multiples of in is the upper half-plane and a fundamental domain of rotations by multiples of is the first quadrant .The concept of a fundamental domain is a generalization of a minimal group block, since while the intersection of fundamental domains has empty interior, the intersection of minimal blocks is the empty set.
A symmetric polynomial on variables , ..., (also called a totally symmetric polynomial) is a function that is unchanged by any permutation of its variables. In other words, the symmetric polynomials satisfy(1)where and being an arbitrary permutation of the indices 1, 2, ..., .For fixed , the set of all symmetric polynomials in variables forms an algebra of dimension . The coefficients of a univariate polynomial of degree are algebraically independent symmetric polynomials in the roots of , and thus form a basis for the set of all such symmetric polynomials.There are four common homogeneous bases for the symmetric polynomials, each of which is indexed by a partition (Dumitriu et al. 2004). Letting be the length of , the elementary functions , complete homogeneous functions , and power-sum functions are defined for by(2)(3)(4)and for by(5)where is one of , or . In addition, the monomial functions are defined as(6)where is the set of permutations..
For any ideal in a Dedekind ring, there is an ideal such that(1)where is a principal ideal, (i.e., an ideal of rank 1). Moreover, for a Dedekind ring with a finite ideal class group, there is a finite list of ideals such that this equation may be satisfied for some . The size of this list is known as the class number.Class numbers are usually studied in the context of the orders of number fields. If this order is maximal, then it is the ring of integers of the number field, in which case the class number is equal to the order of the class group of the number field; otherwise it is equal to the order of the Picard group of the nonmaximal order in question.When the class number of a ring of integers in a number field is 1, the ring corresponding to a given ideal has unique factorization and, in a sense, the class number is a measure of the failure of unique factorization in that ring.A finite series giving exactly the class number of a ring is known as a class number formula...
An algebraic equation in variables is an polynomial equation of the formwhere the coefficients are integers (where the exponents are nonnegative integers and the sum is finite).Examples of algebraic equations are given in the following table.curveequationCayley's sexticeight curveline through and plane through , , and unit circleunit sphereThe roots of an algebraic equation in one variable are known as algebraicnumbers.
A quadratic form is said to be positive definite if for . A real quadratic form in variables is positive definite iff its canonical form is(1)A binary quadratic form(2)of two real variables is positive definite if it is for any , therefore if and the binary quadratic form discriminant . A binary quadratic form is positive definite if there exist nonzero and such that(3)(Le Lionnais 1983).The positive definite quadratic form(4)is said to be reduced if , , and if or . Under the action of the general linear group , i.e., under the set of linear transformations of coordinates with integer coefficients and determinant , there exists a unique reduced positive definite binary quadratic form equivalent to any given one.There exists a one-to-one correspondence between the set of reduced quadratic forms with fundamental discriminant and the set of classes of fractional ideals of the unique quadratic field with discriminant . Let be a reduced positive definite..
The binary quadratic form is said to be reduced if the following conditions hold. Let be the discriminant, then 1. If is negative, is reduced if and if whenever or , and is called real. 2. If is positive, is reduced if , and is called imaginary or positive definite. Every imaginary binary quadratic form is equivalent to a unique reduced form and every real binary quadratic form is equivalent to a finite number of reduced forms.
A totally imaginary field is a field with no real embeddings. A general number field of degree has real embeddings and imaginary embeddings (), where . If , is totally imaginary; if , it is totally real; otherwise it is imaginary but not totally imaginary.
The Chebotarev density theorem is a complicated theorem in algebraic number theory which yields an asymptotic formula for the density of prime ideals of a number field that split in a certain way in an algebraic extension of . When the base field is the field of rational numbers, the theorem becomes much simpler.Let be a monic irreducible polynomial of degree with integer coefficients with root , let , let be the normal closure of , and let be a partition of , i.e., an ordered set of positive integers with . A prime is said to be unramified (over the number field ) if it does not divide the discriminant of . Let denote the set of unramified primes. Consider the set of unramified primes for which factors as modulo , where is irreducible modulo and has degree . Also define the density of primes in as follows:Now consider the Galois group of the number field . Since this is a subgroup of the symmetric group , every element of can be represented as a permutation of letters,..
Given a number field , there exists a unique maximal unramified Abelian extension of which contains all other unramified Abelian extensions of . This finite field extension is called the Hilbert class field of . By a theorem of class field theory, the Galois group is isomorphic to the class group of and for every subgroup of , there exists a unique unramified Abelian extension of contained in such that .The degree of over is equal to the class number of .
Given a number field , a Galois extension field , and prime ideals of and of unramified over , there exists a unique element of the Galois group such that for every element of ,(1)where is the norm of the prime ideal in .The symbol is called an Artin symbol. If is an Abelian extension of , the Artin symbol depends only on the prime ideal of lying under , so it may be written as . In this case, the Artin symbol can be generalized as follows. Let be an ideal of with prime factorization(2)Then the Artin symbol is defined by(3)
Let be a number field with real embeddings and imaginary embeddings and let . Then the multiplicative group of units of has the form(1)where is a primitive th root of unity, for the maximal such that there is a primitive th root of unity of . Whenever is quadratic, (unless , in which case , or , in which case ). Thus, is isomorphic to the group . The generators for are called the fundamental units of . Real quadratic number fields and imaginary cubic number fields have just one fundamental unit and imaginary quadratic number fields have no fundamental units. Observe that is the order of the torsion subgroup of and that the are determined up to a change of -basis and up to a multiplication by a root of unity.The fundamental unit of a number field is intimatelyconnected with the regulator.The fundamental units of a field generated by the algebraic number can be computed in the Wolfram Language using NumberFieldFundamentalUnits[a].In a real quadratic field,..
For a Galois extension field of a field , the fundamental theorem of Galois theory states that the subgroups of the Galois group correspond with the subfields of containing . If the subfield corresponds to the subgroup , then the extension field degree of over is the group order of ,(1)(2)Suppose , then and correspond to subgroups and of such that is a subgroup of . Also, is a normal subgroup iff is a Galois extension field. Since any subfield of a separable extension, which the Galois extension field must be, is also separable, is Galois iff is a normal extension of . So normal extensions correspond to normal subgroups. When is normal, then(3)as the quotient group of the group action of on .According to the fundamental theorem, there is a one-one correspondence between subgroups of the Galois group and subfields of containing . For example, for the number field shown above, the only automorphisms of (keeping fixed) are the identity, , , and , so these form..
Let be a field of field characteristic . Then the Frobenius automorphism on is the map which maps to for each element of .
Let be a number field of extension degree over . Then an order of is a subring of the ring of integers of with generators over , including 1.The ring of integers of every number field is an order, known as the maximal order, of . Every order of is contained in the maximal order. If is an algebraic integer in , then is an order of , though it may not be maximal if is greater than 2.