An element of an extension field of a field which is not algebraic over . A transcendental number is a complex number which is transcendental over the field of rational numbers.
The multiplicative subgroup of all elements in the product of the multiplicative groups whose absolute value is 1 at all but finitely many , where is a number field and a field place.
If a field has the property that, if the sets , ..., , ... belong to it, then so do the sets and , then the field is called a Borel field (Papoulis 1984, p. 29).
The ordered pair , where is the number of real embeddings of the number field and is the number of complex-conjugate pairs of embeddings. The degree of the number field is .
If is an algebraic number of degree , then the totality of all expressions that can be constructed from by repeated additions, subtractions, multiplications, and divisions is called a number field (or an algebraic number field) generated by , and is denoted . Formally, a number field is a finite extension of the field of rational numbers.The elements of a number field which are roots of a polynomialwith integer coefficients and leading coefficient 1 are called the algebraicintegers of that field.The coefficients of an algebraic equations such as the quintic equation can be characterized by the groups of their associated number fields. A database of the groups of number field polynomials is maintained by Klüners and Malle. For example, the polynomial is associated with the group of order 20...
A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra. An archaic name for a field is rational domain. The French term for a field is corps and the German word is Körper, both meaning "body." A field with a finite number of members is known as a finite field or Galois field.Because the identity condition is generally required to be different for addition and multiplication, every field must have at least two elements. Examples include the complex numbers (), rational numbers (), and real numbers (), but not the integers (), which form only a ring.It has been proven by Hilbert and Weierstrass that all generalizations of the field concept to triplets of elements are equivalent to the field of complex numbers...
Given a field and an extension field , if is an algebraic element over , the minimal polynomial of over is the unique monic irreducible polynomial such that . It is the generator of the idealof .Any irreducible monic polynomial of has some root in some extension field , so that it is the minimal polynomial of . This arises from the following construction. The quotient ring is a field, since is a maximal ideal, moreover contains . Then is the minimal polynomial of , the residue class of in ., which is also the simple extension field obtained by adding to . Hence, in this case, and the extension field coincides with the extension ring.In general, if is any other algebraic element of any extension field of with the same minimal polynomial , it remains true that , and this field is isomorphic to ...
The degree (or relative degree, or index) of an extension field , denoted , is the dimension of as a vector space over , i.e.,If is finite, then the extension is said to be finite; otherwise, it is said to be infinite.
A field is said to be an extension field (or field extension, or extension), denoted , of a field if is a subfield of . For example, the complex numbers are an extension field of the real numbers, and the real numbers are an extension field of the rational numbers.The extension field degree (or relative degree, or index) of an extension field , denoted , is the dimension of as a vector space over , i.e.,(1)Given a field , there are a couple of ways to define an extension field. If is contained in a larger field, . Then by picking some elements not in , one defines to be the smallest subfield of containing and the . For instance, the rationals can be extended by the complex number , yielding . If there is only one new element, the extension is called a simple extension. The process of adding a new element is called "adjoining."Since elements can be adjoined in any order, it suffices to understand simple extensions. Because is contained in a larger field,..
An extension field of a field that is not algebraic over , i.e., an extension field that has at least one element that is transcendental over .For example, the field of rational functions in the variable is a transcendental extension of since is transcendental over . The field of real numbers is a transcendental extension of the field of rational numbers, since is transcendental over .
A field which is complete with respect to a discrete valuation is called a local field if its field of residue classes is finite. The Hasse principle is one of the chief applications of local field theory. A local field with field characteristic is isomorphic to the field of power series in one variable whose coefficients are in a finite field. A local field of characteristic zero is either the p-adic numbers, or power series in a complex variable.
The Eisenstein units are the Eisenstein integers , , , where(1)(2)
The transcendence degree of , sometimes called the transcendental degree, is one because it is generated by one extra element. In contrast, (which is the same field) also has transcendence degree one because is algebraic over . In general, the transcendence degree of an extension field over a field is the smallest number elements of which are not algebraic over , but needed to generate . If the smallest set of transcendental elements needed to generate is infinite, then the transcendence degree is the cardinal number of that set.For instance, the transcendence degree of over is one. The transcendence degree of over is an infinite cardinal number. There are many open questions about the traditional constants in mathematics, such as the transcendence degree of .
Let be the cube root of unity . Then the Eisenstein primes are Eisenstein integers, i.e., numbers of the form for and integers, such that cannot be written as a product of other Eisenstein integers.The Eisenstein primes with complex modulus are given by , , , , , 2, , , , , , , , , , , , and . The positive Eisenstein primes with zero imaginary part are precisely the ordinary primes that are congruent to 2 (mod 3), i.e., 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, ... (OEIS A003627).In particular, there are three classes of Eisenstein primes (Cox 1989; Wagon 1991, p. 320): 1. . 2. Numbers of the form for , and a prime congruent to 2 (mod 3). 3. Numbers of the form or where is a prime congruent to 1 (mod 3). Since primes of this form always have the form , finding the corresponding and gives and via and . ..
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.
A grand unified theory of mathematics which includes the search for a generalization of Artin reciprocity (known as Langlands reciprocity) to non-Abelian Galois extensions of number fields. In a January 1967 letter to André Weil, Langlands proposed that the mathematics of algebra (Galois representations) and analysis (automorphic forms) are intimately related, and that congruences over finite fields are related to infinite-dimensional representation theory. In particular, Langlands conjectured that the transformations behind general reciprocity laws could be represented by means of matrices (Mackenzie 2000).In 1998, three mathematicians proved Langlands' conjectures for local fields, and in a November 1999 lecture at the Institute for Advanced Study at Princeton University, L. Lafforgue presented a proof of the conjectures for function fields. This leaves only the case of number fields as unresolved (Mackenzie..
The topological completion of a field with respect to the absolute value is the smallest field containing for which all Cauchy sequences or rationals converge.
The Eisenstein integers, sometimes also called the Eisenstein-Jacobi integers (Finch 2003, p. 601), are numbers of the form , where and are normal integers,(1)is one of the roots of , the others being 1 and(2)The sums, differences, and products of Eisenstein integers is another Eisenstein integer.Eisenstein integers are complex numbers that are members of the imaginary quadratic field , which is precisely the ring (Wagon 1991, p. 320). The field of Eisenstein integers has the six units (or roots of unity), namely , , and (Wagon 1991, p. 320; Guy 1994, p. 35).Every nonzero Eisenstein integer has a unique (up to ordering) factorization up to associates, where associates are Eisenstein integers related to the given Eisenstein integer by rotations of multiples of in the complex plane. Specifically, any nonzero Eisenstein integer is uniquely the product of powers of , , and the "positive" Eisenstein primes,..
A technical mathematical object defined in terms of a polynomial ring of variables over a field . Syzygies occur in tensors at rank 5, 7, 8, and all higher ranks, and play a role in restricting the number of independent isotropic tensors. An example of a rank-5 syzygy iswhere is the permutation tensor and is the Kronecker delta.Syzygies can roughly be viewed as an extension of polynomial greatest common divisors to the multivariable case, i.e., they give a method for solving multivariate polynomial Diophantine equationsSyzygies give the polynomials or else show that no such solution exists. The ability to solve linear multivariable polynomial equations allows computation of multivariate ideal operations such intersection, quotient, and a number of other commutative algebra operations.
If a subset of the elements of a field satisfies the field axioms with the same operations of , then is called a subfield of . In a finite field of field order , with a prime, there exists a subfield of field order for every dividing .
The German mathematician Kronecker proved that all the Galois extensions of the rationals Q with Abelian Galois groups are subfields of cyclotomic fields , where is the group of th roots of unity. He then sought to find a similar function whose division values would generate the Abelian extensions of an arbitrary number field. He discovered that the j-function works for imaginary quadratic fields , but the completion of this problem, known as Kronecker's Jugendtraum ("dream of youth"), for more general fields remains one of the great unsolved problems in number theory.
Two elements , of a field , which is an extension field of a field , are called conjugate (over ) if they are both algebraic over and have the same minimal polynomial.Two complex conjugates and () are also conjugate in this more abstract meaning, since they are the roots of the following monic polynomial(1)with real coefficients, which is irreducible since its discriminant is negative, and hence is their common minimal polynomial over the field of real numbers.All primitive th roots of unity are conjugate over since they have the cyclotomic polynomial as their common minimal polynomial. So, for, instance, the primitive fifth roots of unity(2)(3)(4)(5)are all conjugate over . This shows that elements (such as and ) which are not conjugate over a larger field () may be conjugate over a smaller field.The number of conjugates of an algebraic element over is less than or equal to the degree of its minimal polynomial over , and equality holds iff has no multiple..
Let be a complete non-Archimedean valuated field, with valuation ring , and let be a power series with coefficients in . Suppose at least one of the coefficients is nonzero (so that is not identically zero) and the sequence of coefficients converges to 0 with respect to . Then has only finitely many zeros in .
The Jacobian conjecture in the plane, first stated by Keller (1939), states that given a ring map of (the polynomial ring in two variables over the complex numbers ) to itself that fixes and sends , to , respectively, is an automorphism iff the Jacobian is a nonzero element of . The condition can easily shown to be necessary, but proving sufficiency has been an open problem since Keller (1939).The Jacobian conjecture is one of Smale's problems.There have been at least five published incorrect proofs and many incorrect attempts over the years. In November 2004, Hochster (2004) sent an email announcing a new proof by Carolyn Dean. However, this proof unfortunately contained an error as well.
Let be a vector space over a field , and let be a nonempty set. For an appropriately defined affine space , is called the coefficient field.
Given a set of primes, a field is called a class field if it is a maximal normal extension of the rationals which splits all of the primes in , and if is the maximal set of primes split by K. Here the set is defined up to the equivalence relation of allowing a finite number of exceptions.The basic example is the set of primes congruent to 1 (mod 4),The class field for is because every such prime is expressible as the sum of two squares .
A polynomial map , with in a field is called invertible if there exist such that for so that (Becker and Weispfenning 1993, p. 330). Gröbner bases provide a means to decide for given whether or not is invertible.
Chevalley's theorem, also known as the Chevalley-Waring theorem, states that if is a polynomial in , where is a finite field of field characteristic , and the degree of is less than , then the number of zeros of in is equal to 0 (mod ).In the special case that is a homogeneous polynomial, the theorem states that if and is greater than the degree of , then has at least two zeros in .
The extension field of a field is called a splitting field for the polynomial if factors completely into linear factors in and does not factor completely into linear factors over any proper subfield of containing (Dummit and Foote 1998, p. 448).For example, the extension field is the splitting field for since it is the smallest field containing its roots, and . Note that it is also the splitting field for .
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,..
A polynomial is said to split over a field ifwhere and are in . Then the polynomial is said to split into linear factors. For example, splits over the field of complex numbers since .
An imaginary quadratic field is a quadratic field with . Special cases are summarized in the following table.field membersGaussian integerEisenstein integer
Let be an algebraically closed field and let be an ideal in , where is a finite set of indeterminates. Let be such that for any in , if every element of vanishes when evaluated if we set each (), then also vanishes. Then lies in for some . Colloquially, the theory of algebraically closed fields is a complete model.
The name for the set of integers modulo , denoted . If is a prime , then the modulus is a finite field .
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)
A separable extension of a field is one in which every element's algebraic number minimal polynomial does not have multiple roots. In other words, the minimal polynomial of any element is a separable polynomial. For example,(1)is a separable extension since the minimal polynomial of , when , is(2)In fact, in field characteristic zero, every extension is separable, as is any finite extension of a finite field. If all of the algebraic extensions of a field are separable, then is called a perfect field. It is a bit more complicated to describe a field which is not separable. Consider the field of rational functions with coefficients in , infinite in size and characteristic 2 ().(3)and the extension(4)For instance, and . Then is not separable because is the minimum polynomial for , which has one multiple root. Since in characteristic 2,(5)..
A collection of equations satisfies the Hasse principle if, whenever one of the equations has solutions in and all the , then the equations have solutions in the rationals . Examples include the set of equationswith , , and integers, and the set of equationsfor rational. The trivial solution is usually not taken into account when deciding if a collection of homogeneous equations satisfies the Hasse principle. The Hasse principle is sometimes called the local-global principle.
Take a number field and an Abelian extension, then form a prime divisor that is divided by all ramified primes of the extension . Now define a map from the fractional ideals relatively prime to to the Galois group of that sends an ideal to . This map is called the Artin map. Its importance lies in the kernel, which Artin's reciprocity theorem states contains all fractional ideals that are only composed of primes that split completely in the extension .This is the reason that it is a reciprocity law. The inertia degree of a prime can now be computed since the smallest exponent for which belongs to this kernel, which is exactly the inertia degree, is now known. Now because is unramified and is Galois, , with the inertia degree and the number of factors into which splits when it is extended to . So it is completely known how behaves when it is extended to .This is completely analogous to quadratic reciprocity because it also determines when an unramified prime splits..
A global field is either a number field, a function field on an algebraic curve, or an extension of transcendence degree one over a finite field. From a modern point of view, a global field may refer to a function field on a complex algebraic curve as well as one over a finite field. A global field contains a canonical subring, either the algebraic integers or the polynomials. By choosing a prime ideal in its subring, a global field can be topologically completed to give a local field. For example, the rational numbers are a global field. By choosing a prime number , the rationals can be completed in the p-adic norm to form the p-adic numbers .A global field is called global because of the special case of a complex algebraic curve, for which the field consists of global functions (i.e., functions that are defined everywhere). These functions differ from functions defined near a point, whose completion is called a local field. Under favorable conditions,..
A separable algebraic extension of for which every irreducible polynomial in which has a single root in has all its roots in is said to be Galoisian. Galoisian extensions are also called algebraically normal.
The following are equivalent definitions for a Galois extension field (also simply known as a Galois extension) of .1. is the splitting field for a collection of separable polynomials. When is a finite extension, then only one separable polynomial is necessary. 2. The field automorphisms of that fix do not fix any intermediate fields , i.e., . 3. Every irreducible polynomial over which has a root in factors into linear factors in . Also, must be a separable extension. 4. A field automorphism of the algebraic closure of for which must fix . That is to say that must be a field automorphism of fixing . Also, must be a separable extension. A Galois extension has all of the above properties. For example, consider , the rationals adjoined by the imaginary number , over , which is a Galois extension. Note that contains all of the roots of , and is generated by them, so it is the splitting field of . Of course, there are two distinct roots in so it is separable. The only..
An analytic function satisfying the irreducible algebraic equationwith single-valued meromorphic functions in a complex domain is called a -algebroidal function in .
An algebraic integer of the form where is squarefree forms a quadratic field and is denoted . If , the field is called a real quadratic field, and if , it is called an imaginary quadratic field. The integers in are simply called "the" integers. The integers in are called Gaussian integers, and the integers in are called Eisenstein integers. The algebraic integers in an arbitrary quadratic field do not necessarily have unique factorizations. For example, the fields and are not uniquely factorable, since(1)(2)although the above factors are all primes within these fields. All other quadratic fields with are uniquely factorable.Quadratic fields obey the identities(3)(4)and(5)The integers in the real field are of the form , where(6)There are exactly 21 quadratic fields in which there is a Euclidean algorithm, corresponding to for squarefree integers , , , , , 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, and 73 (A048981). This list..
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..
A finite extension of the field of rational functions in the indeterminate , i.e., is a root of a polynomial , where . Function fields are sometimes called algebraic function fields.
The minimal polynomial of an algebraic number is the unique irreducible monic polynomial of smallest degree with rational coefficients such that and whose leading coefficient is 1. The minimal polynomial can be computed using MinimalPolynomial[zeta, var] in the Wolfram Language package AlgebraicNumberFields` .For example, the minimal polynomial of is . In general, the minimal polynomial of , where and is a prime number, is , which is irreducible by Eisenstein's irreducibility criterion. The minimal polynomial of every primitive th root of unity is the cyclotomic polynomial . For example, is the minimal polynomial ofIn general, two algebraic numbers that are complex conjugates have the same minimal polynomial.Considering the extension field as a finite-dimensional vector space over the field of the rational numbers, then multiplication by induces a linear transformation on . The matrix minimal polynomial of , as a linear transformation,..
Let be a field of field characteristic . Then the Frobenius automorphism on is the map which maps to for each element of .
A subfield which is strictly smaller than the fieldin which it is contained.The field of rationals is a proper subfield of the field of real numbers which, in turn, is a proper subfield of ; is actually the biggest proper subfield of , whereas there are infinite sequences of proper subfields between and . Here is one example, constructed by using the th root of 2 for different prime numbers ,Note that all the fields in the sequence are contained in the set of algebraic numbers, which is another proper subfield of .Hence, has infinitely many proper subfields. On the contrary, has none, since any subfield of must contain 0, all integer multiples of 1, and all their quotients (since every field is a division algebra), thus generating all the rational numbers. In particular, is the smallest proper subfield of .For all prime numbers and integers , the prime field is a proper subfield of ...
A finite field is a field with a finite field order (i.e., number of elements), also called a Galois field. The order of a finite field is always a prime or a power of a prime (Birkhoff and Mac Lane 1996). For each prime power, there exists exactly one (with the usual caveat that "exactly one" means "exactly one up to an isomorphism") finite field GF(), often written as in current usage.GF() is called the prime field of order , and is the field of residue classes modulo , where the elements are denoted 0, 1, ..., . in GF() means the same as . Note, however, that in the ring of residues modulo 4, so 2 has no reciprocal, and the ring of residues modulo 4 is distinct from the finite field with four elements. Finite fields are therefore denoted GF(), instead of GF(), where , for clarity.The finite field GF(2) consists of elements 0 and 1 which satisfy the following addition and multiplication tables.0100111001000101If a subset of the elements..
Given algebraic numbers , ..., it is always possible to find a single algebraic number such that each of , ..., can be expressed as a polynomial in with rational coefficients. The number is then called a primitive element of the extension field . Stated differently, an algebraic number is a primitive element of iff . Primitive elements were implemented in version of the Wolfram Language prior to 6 as PrimitiveElement[z, a1, ..., an] (after loading the package NumberTheory`PrimitiveElement`.For example, a primitive element of is given by , with (1)(2)
A place of a number field is an isomorphism class of field maps onto a dense subfield of a nondiscrete locally compact field .In the function field case, let be a function field of algebraic functions of one variable over a field . Then by a place in , we mean a subset of which is the ideal of nonunits of some valuation ring over .
The prime subfield of a field is the subfield of generated by the multiplicative identity of . It is isomorphic to either (if the field characteristic is 0), or the finite field (if the field characteristic is ).
An extension of a field is said to be algebraic if every element of is algebraic over (i.e., is the root of a nonzero polynomial with coefficients in ).
Given a field and an extension field , an element is called algebraic over if it is a root of some nonzero polynomial with coefficients in .Obviously, every element of is algebraic over . Moreover, the sum, difference, product, and quotient of algebraic elements are again algebraic. It follows that the simple extension field is an algebraic extension of iff is algebraic over .The imaginary unit i is algebraic over the field of real numbers since it is a root of the polynomial . Because its coefficients are integers, it is even true that is algebraic over the field of rational numbers, i.e., it is an algebraic number (and also an algebraic integer). As a consequence, and are algebraic extensions of and respectively. (Here, is the complex field , whereas is the total ring of fractions of the ring of Gaussian integers .)..
A map defined by one or more polynomials. Given a field , a polynomial map is a map such that for all points ,for suitable polynomials . The zero set of is the set of all solutions of the simultaneous equations , and is an algebraic variety in .An example of polynomial map is the th coordinate map , defined by for all . In the language of set theory, it is the projection of the Cartesian product onto the th factor.Polynomial maps can be defined on any nonempty subset of . If is an affine variety, then the set of all polynomial maps from to is the coordinate ring of . If is an affine variety of , then every polynomial map induces a ring homomorphism , defined by . Conversely, every ring homomorphism determines a polynomial map , where .A polynomial map is a real-valued polynomial function. Its graph is the plane algebraic curve with Cartesian equation ...
The field is called an algebraic closure of if is algebraic over and if every polynomial splits completely over , so that can be said to contain all the elements that are algebraic over .For example, the field of complex numbers is the algebraic closure of the field of reals .
A perfect field is a field such that every algebraic extension is separable. Any field in field characteristic zero, such as the rationals or the p-adics, or any finite field is a perfect field. More generally, suppose the characteristic exponent of the field is . Then is perfect iff
The order ideal in , the ring of integral laurent polynomials, associated with an Alexander matrix for a knot . Any generator of a principal Alexander ideal is called an Alexander polynomial. Because the Alexander invariant of a tame knot in has a square presentation matrix, its Alexander ideal is principal and it has an Alexander polynomial .
For a field with multiplicative identity 1, consider the numbers , , , etc. Either these numbers are all different, in which case we say that has characteristic 0, or two of them will be equal. In the latter case, it is straightforward to show that, for some number , we have . If is chosen to be as small as possible, then will be a prime, and we say that has characteristic . The characteristic of a field is sometimes denoted .The fields (rationals), (reals), (complex numbers), and the p-adic numbers have characteristic 0. For a prime, the finite field GF() has characteristic .If is a subfield of , then and have the same characteristic.
If is an element of a field over the prime field , then the set of all rational functions of with coefficients in is a field derived from by adjunction of .
An element of an adèle group, sometimes called a repartition in older literature (e.g., Chevalley 1951, p. 25). Adèles arise in both number fields and function fields. The adèles of a number field are the additive subgroups of all elements in , where is the field place, whose absolute value is at all but finitely many s.Let be a function field of algebraic functions of one variable. Then a map which assigns to every field place of an element of such that there are only a finite number of field places for which is called an adèle (Chevalley 1951, p. 1951).
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.
A field automorphism of a field is a bijective map that preserves all of 's algebraic properties, more precisely, it is an isomorphism. For example, complex conjugation is a field automorphism of , the complex numbers, because(1)(2)(3)(4)A field automorphism fixes the smallest field containing 1, which is , the rational numbers, in the case of field characteristic zero.The set of automorphisms of which fix a smaller field forms a group, by composition, called the Galois group, written . For example, take , the rational numbers, and(5)(6)which is an extension of . Then the only automorphism of (fixing ) is , where . It is no accident that and are the roots of . The basic observation is that for any automorphism , any polynomial with coefficients in , and any field element ,(7)So if is a root of , then is also a root of .The rational numbers form a field with no nontrivial automorphisms. Slightly more complicated is the extension of by , the real cube root..
If is an algebraic Galois extension field of such that the Galois group of the extension is Abelian, then is said to be an Abelian extension of .For example,is the field of rational numbers with the square root of two adjoined, a degree-two extension of . Its Galois group has two elements, the nontrivial element sending to , and is Abelian. By contrast, the degree-six extensionis the splitting field of , and is not an Abelian extension of . Indeed, the six automorphisms of , fixing , are defined by the permutations of the three roots of . So the Galois group in this case is the symmetric group on three letters, which is non-Abelian.In an Abelian extension that is a splitting field for a polynomial , the roots of are related. For instance, consider a cyclotomic field, , where is a primitive root and is a prime number. Then the Galois group is the multiplicative group of the cyclic group .A classical theorem in number theory says that an Abelian extension of the..