example of normal extension

Let F=(2). Then the extension F/ is normal because F is clearly the splitting fieldMathworldPlanetmath of the polynomialPlanetmathPlanetmath f(x)=x2-2. Furthermore F/ is a Galois extensionMathworldPlanetmath with Gal(F/)/2.

Now, let 21/4 denote the positive real fourth root of 2 and define K=F(21/4). Then the extension K/F is normal because K is the splitting field of k(x)=x2-2, and as before K/F is a Galois extension with Gal(K/F)/2.

However, the extension K/ is neither normal nor Galois. Indeed, the polynomial g(x)=x4-2 has one root in K (actually two), namely 21/4, and yet g(x) does not split in K into linear factors.


The Galois closure of K over is L=(21/4,i).

Title example of normal extensionMathworldPlanetmath
Canonical name ExampleOfNormalExtension
Date of creation 2013-03-22 14:30:46
Last modified on 2013-03-22 14:30:46
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Example
Classification msc 12F10
Related topic GaloisExtension
Related topic CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois
Related topic NormalIsNotTransitive
Related topic GaloisIsNotTransitive