#
LMS Durham Symposium on Computational Number Theory

24 July - 3 August 2000

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Explicit construction of number field extensions of small degree, with
applications to discriminant counting

#### (Two talks)

Henri COHEN (Bordeaux)

**Abstract:**
After reviewing some results coming from Class Field Theory and
the theory of higher ramification groups, we show how to construct
explicitly extensions of number fields of type C_l, D_l, A_4 and
S_4 (l prime). We deduce from this both asymptotic and exact
formulas for the number of such extensions. For absolute extensions,
we explain some elementary number-theoretical methods which have
allowed us to count the number of desired extensions up to very high
discriminant (10^{37} in the best case C_3).

This is joint work with F. Diaz y Diaz and M. Olivier.

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