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- W2764042643 abstract "We study solutions to the Brauer embedding problem with restricted ramification. Suppose $G$ and $A$ are a abelian groups, $E$ is a central extension of $G$ by $A$, and $f:text{Gal}(overline{mathbf{Q}}/mathbf{Q})rightarrow G$ a continuous homomorphism. We determine conditions on the discriminant of $f$ that are equivalent to the existence of an unramified lift $widetilde{f}:text{Gal}(overline{mathbf{Q}}/mathbf{Q})rightarrow E$ of $f$. As a consequence of this result, we use conditions on the discriminant of $K$ for $K/mathbf{Q}$ abelian to classify and count unramified nonabelian extensions $L/K$ normal over $mathbf{Q}$ where the (nontrivial) commutator subgroup of $text{Gal}(L/mathbf{Q})$ is contained in its center. This generalizes a result due to Lemmermeyer, which states that a quadratic field $mathbf{Q}(sqrt{d})$ has an unramified extension normal over $mathbf{Q}$ with Galois group $H_8$ the quaternion group if and only if the discriminant factors $d=d_1 d_2 d_3$ as a product of three coprime discriminants, at most one of which is negative, satisfying the following condition on Legendre symbols: [ left(frac{d_i d_j}{p_k}right)=1 ] for ${i,j,k}={1,2,3}$ and $p_i$ any prime dividing $d_i$." @default.
- W2764042643 created "2017-10-20" @default.
- W2764042643 creator A5063777958 @default.
- W2764042643 date "2017-10-02" @default.
- W2764042643 modified "2023-09-27" @default.
- W2764042643 title "Certain Unramified Metabelian Extensions Using Lemmermeyer Factorizations" @default.
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- W2764042643 hasPublicationYear "2017" @default.
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