simple field extension

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

We consider the substitution homomorphismφ:K[X]K[α], where


According to the ring homomorphismMathworldPlanetmath theorem, the image ring K[α] is isomorphic with the residue class ring K[X]/𝔭, where 𝔭 is the ideal of polynomialsMathworldPlanetmathPlanetmathPlanetmath having α as their zero.  Because K[α] is, as subring of the field K(α), an integral domainMathworldPlanetmath, then also K[X]/𝔭 has no zero divisorsMathworldPlanetmath, and hence 𝔭 is a prime idealMathworldPlanetmathPlanetmath.  It must be principal, for K[X] is a principal ideal ring.

There are two possibilities:

  1. 1.

    𝔭=(p(X)), where p(X) is an irreducible polynomialMathworldPlanetmath with  p(α)=0.  Because every non-zero prime ideal of K[X] is maximal, the isomorphic image K[X]/(p(X)) of K[α] is a field, and it must give the of  K(α)=K[α].  We say that α is algebraicMathworldPlanetmath with respect to K (or over K).  In this case, we have a finite field extensionK(α)/K.

  2. 2.

    𝔭=(0).  This means that the homomorphismMathworldPlanetmathPlanetmathPlanetmath φ is an isomorphism between K[X] and K[α], i.e. all expressions aναν behave as the polynomials aνXν.  Now, K[α] is no field because K[X] is not such, but the isomorphy of the rings implies the isomorphy of the corresponding fields of fractionsMathworldPlanetmath.  Thus the simple extension field K(α) is isomorphic with the field K(X) of rational functions in one indeterminate X.  We say that α is transcendental ( with respect to K (or over K).  This time we have a simple infinite field extensionK(α)/K.

Title simple field extension
Canonical name SimpleFieldExtension
Date of creation 2013-03-22 14:23:06
Last modified on 2013-03-22 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