By Ivar Ekeland, Roger Témam

ISBN-10: 204007368X

ISBN-13: 9782040073688

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G/j 5 1 1 let f be its residue degree. Thus ef D ŒF W Qp . K=F / denotes the automorphism group. 1 Two Lemmas In this subsection, we formulate two technical lemmas and describe how they fit together to yield a proof of Theorem 2. First, we focus on the invariants that distinguish between the possible Galois groups for quintic polynomials. One invariant is the discriminant of the polynomial, mentioned previously. If the Galois group G of the polynomial is a subgroup of A5 , then we say the parity of G is C.

X/ D x 2 5. mod 5/. mod 5/ and f is odd or p Qp . 5/ K. This proves parts (2) and (3). t u We note that with a slight modification to the proof of Theorem 2, the case p D 2 can be similarly analyzed. When p D 5, the situation is more complicated, but the details can be extracted from [1]. Absolute Resolvents and Masses of Irreducible Quintic Polynomials 41 Acknowledgements The authors would like to thank the anonymous reviewer for the careful reading and helpful comments. The authors would also like to thank Elon University for supporting this project through internal grants and the Center for Undergraduate Research in Mathematics for their grant support.

Artin and E. Noether, Translated from the seventh German edition by Fred Blum and John R. Schulenberger). MR 1080172 (91h:00009a) A Linear Resolvent for Degree 14 Polynomials Chad Awtrey and Erin Strosnider 1 Introduction Let p be a prime number. An important problem in computational number theory is to determine the Galois group of an irreducible polynomial f defined over the field of p-adic numbers Qp . If the degree of f is either equal to p or is not a multiple of p, then it is straightforward to compute the Galois group of f (see, for example, [1, 10]).

### Analyse convexe et problèmes variationnels (Etudes mathématiques) by Ivar Ekeland, Roger Témam

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