# simple field extension

Let $K(\alpha)$ be obtained from the field $K$ via the of the element $\alpha$, which is called the primitive element of the field extension $K(\alpha)/K$.  We shall settle the of the field $K(\alpha)$.

We consider the substitution homomorphism$\varphi:K[X]\rightarrow K[\alpha]$, where

 $\sum a_{\nu}X^{\nu}\mapsto\sum a_{\nu}\alpha^{\nu}.$

According to the ring homomorphism theorem, the image ring $K[\alpha]$ is isomorphic with the residue class ring $K[X]/\mathfrak{p}$, where $\mathfrak{p}$ is the ideal of polynomials having $\alpha$ as their zero.  Because $K[\alpha]$ is, as subring of the field $K(\alpha)$, an integral domain, then also $K[X]/\mathfrak{p}$ has no zero divisors, and hence $\mathfrak{p}$ is a prime ideal.  It must be principal, for $K[X]$ is a principal ideal ring.

There are two possibilities:

1. 1.

$\mathfrak{p}=(p(X))$, where $p(X)$ is an irreducible polynomial with  $p(\alpha)=0$.  Because every non-zero prime ideal of $K[X]$ is maximal, the isomorphic image $K[X]/(p(X))$ of $K[\alpha]$ is a field, and it must give the of  $K(\alpha)=K[\alpha]$.  We say that $\alpha$ is algebraic with respect to $K$ (or over $K$).  In this case, we have a finite field extension$K(\alpha)/K$.

2. 2.

$\mathfrak{p}=(0)$.  This means that the homomorphism $\varphi$ is an isomorphism between $K[X]$ and $K[\alpha]$, i.e. all expressions $\sum a_{\nu}\alpha^{\nu}$ behave as the polynomials $\sum a_{\nu}X^{\nu}$.  Now, $K[\alpha]$ is no field because $K[X]$ is not such, but the isomorphy of the rings implies the isomorphy of the corresponding fields of fractions.  Thus the simple extension field $K(\alpha)$ is isomorphic with the field $K(X)$ of rational functions in one indeterminate $X$.  We say that $\alpha$ is http://planetmath.org/Algebraic) with respect to $K$ (or over $K$).  This time we have a simple infinite field extension$K(\alpha)/K$.

Title simple field extension SimpleFieldExtension 2013-03-22 14:23:06 2013-03-22 14:23:06 pahio (2872) pahio (2872) 25 pahio (2872) Definition msc 12F99 PrimitiveElementTheorem CanonicalFormOfElementOfNumberField primitive element