S4andS4Extensions of Q Ramified at Only One Prime

✍ Scribed by Darrin Doud

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 161 KB
• Volume: 75
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2345

✦ Synopsis

For each prime p in a certain family of odd primes, we construct an S 4 extension of Q unramified outside p. We show that for all p#3 (mod 8) in our family, this S 4 extension embeds in an S 4 extension, which is also unramified outside p. Invoking Serre's conjecture (in a proven case) allows us to relate the splitting of primes in these extensions to certain modular forms of level 1.

1999 Academic Press

Remarks. Given the conditions on n and k, we see that k must be odd. Also, either n is exactly divisible by 3, or n#1 (mod 3). Any p of this form