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On octahedral extensions of and quadratic -curves

Julio Fernández (2003)

Journal de théorie des nombres de Bordeaux

We give a necessary condition for a surjective representation Gal ( ¯ / ) PGL 2 ( 𝔽 3 ) to arise from the 3 -torsion of a -curve. We pay a special attention to the case of quadratic -curves.

On the compositum of all degree d extensions of a number field

Itamar Gal, Robert Grizzard (2014)

Journal de Théorie des Nombres de Bordeaux

We study the compositum k [ d ] of all degree d extensions of a number field k in a fixed algebraic closure. We show k [ d ] contains all subextensions of degree less than d if and only if d 4 . We prove that for d > 2 there is no bound c = c ( d ) on the degree of elements required to generate finite subextensions of k [ d ] / k . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of d , but that one can take c = d when d is prime. This question was inspired by work of Bombieri and...

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