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 $\phi :K[X]\to K[\alpha ]$, 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 $K[X]/\U0001d52d$, where $\U0001d52d$ 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 $K[X]/\U0001d52d$ has no zero divisors^{}, and hence $\U0001d52d$ is a prime ideal^{}. It must be principal, for $K[X]$ is a principal ideal ring.
There are two possibilities:

1.
$\U0001d52d=(p(X))$, where $p(X)$ is an irreducible polynomial^{} with $p(\alpha )=0$. Because every nonzero 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.
$\U0001d52d=(0)$. This means that the homomorphism^{} $\phi $ 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 transcendental (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 

Canonical name  SimpleFieldExtension 
Date of creation  20130322 14:23:06 
Last modified on  20130322 14:23:06 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  25 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 12F99 
Related topic  PrimitiveElementTheorem 
Related topic  CanonicalFormOfElementOfNumberField 
Defines  primitive element 