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Extension Field -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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A field K is said to be an extension field (or field extension, or extension), denoted K/F, of a field F if F is a subfield of K. 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 K/F, denoted [K:F], is the dimension of K as a vector space over F, i.e.,

[K:F]=dim_FK.

(1)

Given a field F, there are a couple of ways to define an extension field. If F is contained in a larger field, F subset F^'. Then by picking some elements alpha_i in F^' not in F, one defines F(alpha_i) to be the smallest subfield of F^' containing F and the alpha_i. For instance, the rationals can be extended by the complex number zeta, yielding Q(zeta). 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 alpha_i is contained in a larger field, its algebraic operations, such as multiplication and addition, are defined with elements in F. Hence,

F(alpha)={(f(alpha))/(g(alpha)):f,g are polynomials in F and g(alpha)!=0 in F^'}.

(2)

The expression above shows that the polynomials with p(alpha)=0 are important. In fact, there are two possibilities.

1. For some positive integer n, the nth power alpha^n can be written as a (finite) linear combination

alpha^n=sum_(i=0)^(n-1)c_ialpha^i,

(3)

with c_i in F, of powers of alpha less than n. In this case, alpha is called an algebraic number over F and F(alpha) is an algebraic extension. The extension field degree of the extension is the smallest integer n satisfying the above, and the polynomial p(x)=x^n-sum_(i=0)^(n-1) is called the extension field minimal polynomial.

2. Otherwise, there is no such integer n as in the first case. Then alpha is a transcendental number over F and F(alpha) is a transcendental extension of transcendence degree 1.

Note that in the case of an algebraic extension (case 1 above), the extension field can be written

F(alpha)=F[alpha]={f(alpha): f is a polynomial in F and degf<n}.

(4)

Unlike the similar expression above, it is not immediately obvious that the ring F[alpha] is a field. The following argument shows how to divide in this ring. Because no polynomial f of degree less than n can divide the extension field minimal polynomial p, any such polynomial f is relatively prime. That is, there exist polynomials a and b such that af+bp=1, or rather,

a(x)f(x)=1 (mod p(x))

(5)

and a(alpha) is the multiplicative inverse of f(alpha).

Another method for defining an extension is to use an indeterminate variable x. Then F(x) is the set of rational functions in one variable with coefficients in F, and up to isomorphism is the unique transcendental extension of transcendence degree 1. The polynomials F[x] are the denominators and numerators of the rational functions. Given a nonconstant polynomial p(x) which is irreducible over F, the quotient ring F[x]/(p) are the polynomials mod p. In particular, as in case 1 above, F[x]/(p) is generated by 1,x,...,x^(n-1) where n is the degree of p. The field of fractions of F[x]/(p), written F(x)/(p), is an algebraic extension of F, which is isomorphic to the extension of F by one of the roots of p. For instance, Q(i)=Q(x)/(x^2+1). Consequently, if alpha and beta are different roots of an irreducible polynomial p, then F(alpha)=F(beta). When beta in alpha, this isomorphism reflects a field automorphism, one of the symmetries of the field that form the Galois group.

A number field is a finite algebraic extension of the rational numbers. Mathematicians have been using number fields for hundreds of years to solve equations like x^2-2y^2=k where all the variables are integers, because they try to factor the equation in the extension Q(sqrt(2)). For instance, it is easy to see that the only integer solutions to x^2-y^2=(x+y)(x-y)=5 are (+/-3,+/-2) since there are four ways to write 5 as the product of integers.

5=5×1=1×5=-1×-5=-5×-1.

(6)

Hence, it became necessary to understand what is a prime number in a number field. In fact, it led to some confusion because unique factorization does not always hold. The lack of unique factorization is measured by the class group, and the class number.

It can be shown that any number field can be written Q(zeta) for some zeta, that is every number field is a simple extension of the rationals. Naturally, the choice of zeta is not unique, e.g., Q(zeta)=Q(2+zeta)=Q(-zeta)=....

See also

Class Group, Class Number, Extension Field Degree, Extension Ring, Field, Field Automorphism, Galois Theory, Pythagorean Extension, Simple Extension, Splitting Field, Subfield

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Extension Field." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExtensionField.html

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