Arithmetic Lifting of Dihedral Extensions

✍ Scribed by Elena V Black

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 220 KB
• Volume: 203
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7466

✦ Synopsis

The Inverse Galois problem remains a fascinating yet unanswered question. The standard approach through algebraic geometry is to construct a Galois branched covering of the projective line over the rationals with a desired group G. Then one invokes the Hilbert Irreducibility Theorem to construct a G-Galois extension of ‫.ޑ‬ If every Galois extension of number fields is the specialization of a Galois branched covering with the same group, this approach is very logical. Of course, the question whether every Galois extension of number fields is the specialization of a branched covering is of interest in its own right. Answered affirmatively, it would put all Galois extensions of a given number field in families. It is conjectured that the answer to this question is yes. Beckmann addressed this problem in her paper ''Is every extension of ‫ޑ‬ the specialization of a w x branched covering?'' 1 . She answers it affirmatively when G is either a symmetric or a finite abelian group. We will take one step further and prove this conjecture when G is a dihedral group D with n odd. n However, before proving this result, we will reexamine the case of an abelian group. In Section I, we reinterpret the work of Beckmann and Saltman in terms of etale cohomology. In Section II, we generalize to the ćase of a dihedral group.