Chapter 5 Fields



Section 5.1 Extension Fields

In this section we shall be concerned with the relation of one field to another. Let be a field; a field is said to be an extension of if contains . Equivalently, is an extension of if is a subfield of . Throughout this chapter will denote a given field and an extension of .

As was pointed out earlier, in the chapter on vector spaces, if is an extension of , then, under the ordinary field operations in is a vector space over . As a vector space we may talk about linear dependence, dimension, bases, etc., in relative to .

DEFINITION: The degree of over is the dimension of as a vector space over .

We shall always denote the degree of over by . Of particular interest to us is the case in which is finite, that is, when is finite-dimensional as a vector space over . This situation is described by saying that is a finite extension of .

We start off with a relatively simple but, at the same time, highly effective result about finite extensions, namely,

THEOREM 5.1.1: If is a finite extension of and if is a finite extension of then is a finite extension of F. Moreover,



Suppose that are three fields in the relation and, suppose further that is finite. Clearly, any elements in linearly independent over are, all the more so, linearly independent over . Thus the assumption that is finite forces the conclusion that is finite. Also, since is a subspace of is finite. By the theorem, , whence . We have proved the

COROLLARY: If is a finite extension of and is a subfield of which contains , then



Thus, for instance, if is a prime number, then there can be no fields properly between and . A little later, in Section 5.4, when we discuss the construction of certain geometric figures by straightedge and compass, this corollary will be of great significance.

DEFINITION: An element is said to be algebraic over if there exist elements in , not all , such that

.

If the polynomial the ring of polynomials in over , and if

,

then for any element , by we shall mean the element in . In the expression commonly used, is the value of the polynomial obtained by substituting for . The element is said to satisfy if . In these terms is algebraic over if there is a nonzero polynomial which satisfies, that is, for which .

Let be an extension of and let be in . Let be the collection of all subfields of which contain both and is not empty, for itself is an element of . Now, as is easily proved, the intersection of any number of subfields of is again a subfield of . Thus the intersection of all those subfields of which are members of is a subfield of . We denote this subfield by . What are its properties? Certainly it contains both and , since this is true for every subfield of which is a member of . Moreover, by the very definition of intersection, every subfield of in contains , yet itself is in . Thus is the smallest subfield of containing both and . We call the subfield obtained by adjoining to .

Our description of , so far, has been purely an external one. We now give an alternate and more constructive description of . Consider all these elements in which can be expressed in the form here the 's can range freely over and can be any nonnegative integer. As elements in , one such element can be divided by another, provided the latter is not . Let be the set of all such quotients. We leave it as an exercise to prove that is a subfield of .

On one hand, certainly contains and , whence . On the other hand, any subfield of which contains both and , by virtue of closure under addition and multiplication, must contain all the elements



where each . Thus must contain all these elements; being a subfield of must also contain all quotients of such elements. Therefore, . The two relations of course imply that . In this way we have obtained an internal construction of , namely as .

We now intertwine the property that is algebraic over with macroscopic properties of the field itself. This is

THEOREM 5.1.2: The element is algebraic over if and only if is a finite extension of .


DEFINITION: The element is said to be algebraic of degree over if it satisfies a nonzero polynomial over of degree but no nonzero polynomial of lower degree.

In the course of proving Theorem 5.1.2 (in each proof we gave), we proved a somewhat sharper result than that stated in that theorem, namely,

THEOREM 5.1.3: If is algebraic of degree over then

.

This result adapts itself to many uses. We give now, as an immediate consequence thereof, the very interesting

THEOREM 5.1.4: If in are algebraic over then , , and (if ) are all algebraic over . In other words, the elements in which are algebraic over form a subfield of .


Here, too, we have proved somewhat more. Since , every element in satisfies a polynomial of degree at most over , whence the

COROLLARY: If and in are algebraic over of degrees and respectively, then , , and (if ) are algebraic over of degree at most .

In the proof of the last theorem we made two extensions of the field . The first we called ; it was merely the field . The second we called and it was . Thus

;

it is customary to write it as . Similarly, we could speak about ; it is not too difficult to prove that . Continuing this pattern, we can define for elements in .

DEFINITION: The extension of is called an algebraic extension of if every element in is algebraic over .

We prove one more result along the lines of the theorems we have proved so far.

THEOREM 5.1.5: If is an algebraic extension of and if is an algebraic extension of then is an algebraic extension of .


A quick description of Theorem 5.1.5: algebraic over algebraic is algebraic.

The preceding results are of special interest in the particular case in which is the field of rational numbers and the field of complex numbers.

DEFINITION: A complex number is said to be an algebraic number if it is algebraic over the field of rational numbers.

A complex number which is not algebraic is called transcendental. At the present stage we have no reason to suppose that there are any transcendental numbers. In the next section we shall prove that the familiar real number is transcendental. This will, of course, establish the existence of transcendental numbers. In actual fact, they exist in great abundance; in a very well-defined way there are more of them than there are algebraic numbers.

Theorem 5.1.4 applied to algebraic numbers proves the interesting fact that {\it the algebraic numbers form a field}; that is, the sum, products, and quotients of algebraic numbers are again algebraic numbers.

Theorem 5.1.5 when used in conjunction with the so-called ``fundamental theorem of algebra,'' has the implication that the roots of a polynomial whose coefficients are algebraic numbers are themselves algebraic number.