example of an extension that is not normal

In this entry, 23 indicates the real cube root of 2.

Consider the extension (23)/. The minimal polynomial for 23 over is x3-2. This polynomialMathworldPlanetmathPlanetmathPlanetmath factors in (23) as x3-2=(x-23)(x2+x23+43). Let f(x)=x2+x23+43. Note that disc(f(x))=(23)2-443=43-443=-343<0. Thus, f(x) has no real roots. Therefore, f(x) has no roots in (23) since (23). Hence, x3-2 has a root in (23) but does not split in (23). It follows that the extension (23)/ is not normal.

Title example of an extension that is not normal
Canonical name ExampleOfAnExtensionThatIsNotNormal
Date of creation 2013-03-22 16:00:28
Last modified on 2013-03-22 16:00:28
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 6
Author Wkbj79 (1863)
Entry type Example
Classification msc 12F10