ALGEBRAIC EXTENSIONS

The field extensions with which we shall be concerned throughout most of this book are algebraic extensions. These are the field extensions defined by the condition that every element of the top field should be a root of a nontrivial polynomial equation with coefficients in the bottom field. In this section, algebraic elements and extensions will be defined, and their basic properties will be discussed in detail. These properties constitute the foundation of our subject, and are essential for the understanding of every subsequent section.

Let K be a. field, and let L be an extension field of K. We recall that for each α ∈ L, the mapping f(X) → f(α) from K[X] to L is a K-homomorphism having as its image the subdomain K[α] of L; and that its kernel, which is the ideal of K[X] consisting of all polynomials in K[X] admitting α as a zero, is said to be the ideal of algebraic relations of α over K.

We say that an element of L is algebraic overK or transcendental overK according as its ideal of algebraic relations over K is nonnull or null. Equivalently, we could say that an element of L is algebraic over K or transcendental over K according as it is or is not a zero of a nonzero polynomial in K[X].