 Monday 04 Apr 2016: Some problems on cyclic extensions

James Quah - Singapore

H102 14:30-15:30

1. Suppose you want to construct a cyclic cubic field. Given a conductor (some number divisible only by primes congruent to 1 mod 3, union {9}), how do you find all fields of that conductor? (Generalises to odd primes.)

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2. Given an imaginary quadratic field, how do you find all cyclic cubic extensions of it? This is not very difficult, but with a lot more difficulty, I have found the construction for primes 5 and 7.

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I have found a nice congruence formula to find the primes that split in each of the two cubics with conductor 7*13, for example. And also a way to find the minimal polynomial. This must be known, but isn't found on the internet.

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When finding extensions of imaginary quadratics, the problem is quite interesting, and there are both the computational and conceptual approaches. The computational approach is to obtain explicit polynomials of degree 3/5/7 with Galois group D6/D10/D14 and subfield say Q(i). Conceptually, these are subfields of torsion point fields of the elliptic curve y^2 = x^3 + x. For Q(sqrt(-3)), the cubics are equivalent to Kummer extensions.Ā

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