Extension Field

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, its algebraic operations, such as multiplication and addition, are defined with elements in . Hence,

(2)

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

(3)

with , of powers of less than . In this case, is called an algebraic number 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 a transcendental number 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

(4)

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 no polynomial of degree less than can divide the extension field minimal polynomial , any such polynomial is relatively prime. That is, there exist polynomials and such that , or rather,

(5)

and is the multiplicative inverse of .

Another method for defining an extension is to use an indeterminate variable . Then is the set of rational functions in one variable with coefficients in , and up to isomorphism is the unique transcendental extension of transcendence degree 1. The polynomials are the denominators and numerators of the rational functions. Given a nonconstant polynomial which is irreducible over , the quotient ring are the polynomials mod p. In particular, as in case 1 above, is generated by where is the degree of . The field of fractions of , written , is an algebraic extension of , which is isomorphic to the extension of by one of the roots of . For instance, . Consequently, if and are different 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 algebraic 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.

(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 for some , that is every number field is a simple extension of the rationals. Naturally, the choice of is not unique, e.g., .