Friday, February 17, 2017

galois theory - Degree 4 extension of mathbbQ with no intermediate field




I am looking for a degree 4 extension of Q with no intermediate field. I know such extension is not Galois (equivalently not normal). So I was trying to adjoin a root of an irreducible quartic. But I got stuck. Any hint/idea/solution?


Answer



With regard to Sebastian Schoennenbeck's comment, an extension of Q with Galois group A4 (alternating group on 4 points) will do the trick.
Such an extension certainly exists, in fact all alternating groups are Galois groups over Q.


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