An exponential sum of the form(1)where is a real polynomial (Weyl 1914, 1916; Montgomery 2001). Writing(2)a notation introduced by Vinogradov, Weyl observed that(3)(4)(5)(6)a process known as Weyl differencing (Montgomery 2001).Weyl was able to use this process to show that if(7)is a real polynomial and at least one of , ..., is irrational, then is uniformly distributed (mod 1).
The study of number fields by embedding them in a local field is called local class field theory. Information about an equation in a local field may give information about the equation in a global field, such as the rational numbers or a number field (e.g., the Hasse principle).Local class field theory is termed "local" because the local fields are localized at a prime ideal in the ring of algebraic integers. The methods of using class fields have developed over the years, from the Legendre symbol, to the group characters of Abelian extensions of a number field, and is applied to local fields.
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.
If is a root of a nonzero polynomial equation(1)where the s are integers (or equivalently, rational numbers) and satisfies no similar equation of degree , then is said to be an algebraic number of degree .A number that is not algebraic is said to be transcendental. If is an algebraic number and , then it is called an algebraic integer.In general, algebraic numbers are complex, but they may also be real. An example of a complex algebraic number is , and an example of a real algebraic number is , both of which are of degree 2.The set of algebraic numbers is denoted (Wolfram Language), or sometimes (Nesterenko 1999), and is implemented in the Wolfram Language as Algebraics.A number can then be tested to see if it is algebraic in the Wolfram Language using the command Element[x, Algebraics]. Algebraic numbers are represented in the Wolfram Language as indexed polynomial roots by the symbol Root[f, n], where is a number from 1 to the degree of the polynomial..
If is a root of the polynomial equationwhere the s are integers and satisfies no similar equation of degree , then is called an algebraic integer of degree . An algebraic integer is a special case of an algebraic number (for which the leading coefficient need not equal 1). Radical integers are a subring of the algebraic integers.A sum or product of algebraic integers is again an algebraic integer. However, Abel's impossibility theorem shows that there are algebraic integers of degree which are not expressible in terms of addition, subtraction, multiplication, division, and root extraction (the elementary operations) on rational numbers. In fact, if elementary operations are allowed on real numbers only, then there are real numbers which are algebraic integers of degree 3 that cannot be so expressed.The Gaussian integers are algebraic integers of , since are roots of..