Integral Galois Module Structure of Some Lubin–Tate Extensions

✍ Scribed by Nigel P. Byott

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 259 KB
• Volume: 77
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2385

✦ Synopsis

Let K be a finite extension of Q p , and suppose that KÂQ p is ramified and that the residue field of K has cardinality at least 3. Let K (2) be the second division field of K with respect to a Lubin Tate formal group, and let 1 =Gal(K (2) ÂK). We determine the associated order in K1 of the valuation ring O (2) of K (2) , and show that O (2) is not free over this order. The integral Galois module structure of certain intermediate fields E of K (2) ÂK is also considered. In particular, if p{2 and K has residue field of cardinality p or p 2 , we show that the valuation ring of E is free over its associated order if and only if EÂK is either tamely ramified or a p-extension. We also prove that the valuation ring of any weakly ramified abelian extension of K is free over its associated order.