proof of casus irreducibilis for real fields
The classical statement of the casus irreducibilis is that if is an irreducible cubic polynomial with rational coefficients and three real roots, then the roots of are not expressible using real radicals. One example of such a polynomial is , whose roots are .
Proof. That is obvious, and since is radical, and is real since . implies that has order a power of . Since is a -group, it has a nontrivial center (this follows directly from the class equation, or look here (http://planetmath.org/ANontrivialNormalSubgroupOfAFinitePGroupGAndTheCenterOfGHaveNontrivialIntersection)) and thus has a normal subgroup of order , which corresponds to a subfield of Galois over with . But then is also a -group, so inductively we see that we can write
where . Thus each is obtained from by adjoining a square root; it must be a real square root since . This shows that and .
The meat of the proof is in showing that . Let the roots of be , and assume, by renumbering if necessary, that lies in a real radical extension of but that is not a power of . Choose an odd prime dividing , and choose an element of order . Then is not the identity, so for some , . Also, since is irreducible, acts transitively on the roots of , so for some , . Then does not fix , since
Let be the fixed field of . Then is Galois over , and clearly . But Galois subfields of real radical extensions are at most quadratic, so cannot lie in a real radical extension of .
However, , and is prime. Thus (since ). Additionally, since is a real radical extension of , we have also that is a real radical extension of . So lies in the real radical extension of . But this is a contradiction and thus must be a power of .
One consequence of this theorem is the fact that if has degree not a power of , then if has all real roots, those roots are not expressible in terms of real radicals. If , we recover the original casus irreducibilis.
- 1 D.A. Cox, Galois Theory, Wiley-Interscience, 2004.
|Title||proof of casus irreducibilis for real fields|
|Date of creation||2013-03-22 17:43:08|
|Last modified on||2013-03-22 17:43:08|
|Last modified by||rm50 (10146)|