- #1
coquelicot
- 299
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Two days ago, I was absolutely certain to have proved the theorem hereafter. But then, micromass pointed out that another theorem the truth of which I was also certain was in fact false. It seems that in mathematics, never be certain until other mathematicians are.
This is the reason why I would appreciate some comments about the following theorem, and also, if a reliable source can be indicated (I imagine that something like this has already been proved):
Let f be an irreducible polynomial over a field K, and assume that g divides f in some extension of K. Let M be the splitting field of f over K, and L be the field generated by the coefficients of g over K (so K<L<M). Then the product of the elements of the set of the distinct conjugates g^s of g, where s belong to Gal(M/K), is equal to cf^n, with n=[L:K]deg(g)/deg(f) and c in K.
My proof can be found in
https://upload.wikimedia.org/wikipedia/commons/d/de/Bensimhoun-1.lemma_in_Galois_Theory-2.RxInterQuotR-3.conjugates_of_polynomial.pdf , pp. 5-6 (thm. 3.1).
thx.
This is the reason why I would appreciate some comments about the following theorem, and also, if a reliable source can be indicated (I imagine that something like this has already been proved):
Let f be an irreducible polynomial over a field K, and assume that g divides f in some extension of K. Let M be the splitting field of f over K, and L be the field generated by the coefficients of g over K (so K<L<M). Then the product of the elements of the set of the distinct conjugates g^s of g, where s belong to Gal(M/K), is equal to cf^n, with n=[L:K]deg(g)/deg(f) and c in K.
My proof can be found in
https://upload.wikimedia.org/wikipedia/commons/d/de/Bensimhoun-1.lemma_in_Galois_Theory-2.RxInterQuotR-3.conjugates_of_polynomial.pdf , pp. 5-6 (thm. 3.1).
thx.
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