𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Trinomial extensions of Q with ramification conditions

✍ Scribed by Bernat Plans; Núria Vila

Publisher
Elsevier Science
Year
2004
Tongue
English
Weight
250 KB
Volume
105
Category
Article
ISSN
0022-314X

No coin nor oath required. For personal study only.

✦ Synopsis

This paper concerns trinomial extensions of Q with prescribed ramification behavior. We first characterize the positive integers n such that, for every finite set S of prime numbers, there exists a degree n monic trinomial in Z½X whose Galois group over Q is contained in the alternating group A n and such that its discriminant is not divisible by any prime p in S: We also characterize the positive integers n such that, for a given finite set of primes S; there exist trinomial extensions with Galois group over Q contained in A n which are not ramified at the primes of S: In addition, we study the existence of trinomial extensions of Q with Galois group A n which are tamely ramified. In particular, we show that such extensions do exist for every odd n: On the other hand, we obtain that, for n 4 ðmod 8Þ; every A n -extension of Q defined by a degree n trinomial must be wildly ramified at p ¼ 2:

📜 SIMILAR VOLUMES

On the Zℓ-Rank of Abelian Extensions wit
On the Zℓ-Rank of Abelian Extensions with Restricted Ramification
✍ Christian Maire 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 219 KB

For a prime number a, we consider a-extensions k S of number fields k unramified outside a finite set S … S a of places of k, and we study the Z a -rank of the abelian part of k S /k and the number of relations of the Galois group G S :=Gal(k S /k).