On the Stickelberger Ideal and Circular Units of a Compositum of Quadratic Fields

On the Parity of the Class Number of the Field Q(p,q,r)

“…Let C be the group generated by &1 and by all conjugates of = n S , where S [p, q, r]. Theorem 1 of [2] states that [&1, = p , = q , = r , = pq , = pr , = qr , = pqr ] is a basis of C, and that…”

“…In this paragraph we show that such ' can exist only if at least one of x pq , x pr , x qr , x pqr is nonzero. We will use the next statement taken from [2]: Lemma 2.1. In the notation of the beginning of this section let S J finite and l # S. Then…”

On a conjecture concerning minus parts in the style of Gross

“…It is easy to see that our Θ U is equal to e − θ f m U (−1) in the notation of Sinnott (see [10]). Using well-known norm relations (see Lemma 12 in [9] for example) we obtain…”

Sur la structure galoisienne des unités circulaires dans les Zp-extensions

“…On choisit de pre senter ici un exemple plus significatif, avec p=3, mais un peu plus difficile aÁ de crire explicitement. Pour les probleÁ mes survenant en 2 on pourra se rapporter aÁ [10] ouÁ une ame lioration explicite en 2 de C S (K) est construite. Les calculs ne cessaires aÁ l'e laboration de cet exemple ont e te re alise s aÁ l'aide du logiciel GP PARI.…”