Explicit Constructions of the non-Abelian $\mathbf{p^3}$-Extensions Over $\mathbf{\QQ}$

Details Explicit Constructions of the non-Abelian $\mathbf{p^3}$-Extensions Over $\mathbf{\QQ}$ Explicit Constructions of the non-Abelian $\mathbf{p^3}$-Extensions Over $\mathbf{\QQ}$

2008-12-11

arxiv.org/abs/0812.2167v1

Mathematics

Number Theory

12 pages. keywords: Constructive Galois Theory, Heisenberg group, Explicit Embedding problem, Minimal Ramification

Scientific paper

Let $p$ be an odd prime. Let $F/k$ be a cyclic extension of degree $p$ and of characteristic different from $p$. The explicit constructions of the non-abelian $p^{3}$-extensions over $k$, are induced by certain elements in ${F(\mu_{p})}^{*}$. In this paper we let $k=\QQ$ and present sufficient conditions for these elements to be suitable for the constructions. Polynomials for the non-abelian groups of order 27 over $\QQ$ are constructed. We describe explicit realizations of those groups with exactly two ramified primes, without consider Scholz conditions.

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