An extension field (or field extension, or extension) , denoted , of a field is a field for which 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.,

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."

A finitely generated extension can be constructed by successively adjoining single elements, so its construction reduces to a sequence of simple extensions. Because is contained in a larger field, its algebraic operations, such as multiplication and addition, are defined with elements in . Hence,

The expression above shows that the polynomials with are important. In fact, there are two possibilities.

1. For some positive integer , the th power can be written as a (finite) linear combination

with , of powers of less than . In this case, is called algebraic over and is an algebraic extension. The extension field degree of the extension is the smallest integer satisfying the above, and the polynomial is called the extension field minimal polynomial.

2. Otherwise, there is no such integer as in the first case. Then is transcendental over and 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

Unlike the similar expression above, it is not immediately obvious that the ring is a field. The following argument shows how to divide in this ring. Because the extension field minimal polynomial is an irreducible polynomial, every nonzero polynomial of polynomial degree less than is relatively prime to . That is, there exist polynomials and such that , or rather,

and is the multiplicative inverse of .

Another method for defining an extension is to use an indeterminate variable . Then is the field of rational functions in one variable with coefficients in . Up to -isomorphism, it is the simple transcendental extension obtained by adjoining one transcendental element; its transcendence degree is 1. Other extensions of transcendence degree 1 need not be isomorphic to . The polynomials are the denominators and numerators of the rational functions. Given a nonconstant polynomial which is irreducible over , the quotient ring is the ring of polynomials modulo . In particular, as in case 1 above, is generated by where is the polynomial degree of . The quotient ring is already a field and is an algebraic extension of , isomorphic to the extension obtained by adjoining a polynomial root of . For instance, . Consequently, if and are different polynomial roots of an irreducible polynomial , then . When , this isomorphism reflects a field automorphism, one of the symmetries of the field that form the Galois group.

A number field is a finite extension of the rational numbers. Mathematicians have been using number fields for hundreds of years to solve equations like where all the variables are integers, because they try to factor the equation in the extension . For instance, it is easy to see that the only integer solutions to are since there are four ways to write 5 as the product of integers.

Hence, it became necessary to understand what is a prime element in the ring of integers of a number field. In fact, it led to some confusion because unique factorization does not always hold in that ring. 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 for some , that is every number field is a simple extension of the rationals. Naturally, the choice of is not unique, e.g., .

See also

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

Portions of this entry contributed by Todd Rowland

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Cite this as:

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

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