##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Malcolm Bovey

#### Graduate Student, King's College London

## A congruence for S-units of a totally real field

##### Abstract:

Let $p$ be a prime number, $K$ a $CM$ field containing a primitive $p^(n)th$ root of unity and k the maximal real subfield of $K$. We show that a special case of a conjecture of Solomon gives rise to a (conjectural) congruence mod $p^n$, relating certain $S$-units of $k$ to the principal semi-local units of $K$, via the use of local Hilbert symbols. We will discuss the progress made in proving this conjecture and (briefly) some computational verifications.

### December 7, 2006

### 1:00 PM

### AP&M 7218

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