Multiplicative Groups of Galois Extensions

✍ Scribed by W.J. Haboush

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 544 KB
• Volume: 165
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1994.1101

✦ Synopsis

Suppose that (K) is Galois over (k) with group (G), and suppose that (F_{1} \cdots F_{n}) are maximal among the intermediate subfields. Then it is shown that if (G=D_{p}, p) an odd prime, then (K^{} / F_{1}^{} \cdots F_{n}^{}) is a subgroup of (F^{} / k^{} \cdot\left(F^{}\right)^{p}) where (F) is the unique proper Galois subfield. One then deduces that if (G) contains two dihedral groups (D_{p}) and (D_{q}, p \neq q) and both odd, then (K^{}=F_{1}^{} \ldots F_{n}^{*}). These results are derived from calculations involving modules over the integral group ring (\mathbb{Z}[G] . \quad 1994) Academic Press, Inc.