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Questions about the branch of algebra that deals with groups.

0 votes
Accepted

Same rational points

No: Take $k$ to be the rational numbers and $G$ to be the group of third roots of unity. Then the only rational point in $G$ is $1$. Then take $H$ to be the component of the identity. This satisfies y …
Daniel Loughran's user avatar
3 votes
Accepted

Does viewing multiplicative functions as morphisms from $\mathbb{Q}^*\to\mathbb{C}^*$ have a...

Actually number theorists more-or-less do the exact opposite of what you suggest. Namely, as explained in the comments, a multiplicative function is determined by its values on the primes. This sugges …
Daniel Loughran's user avatar
11 votes

Which groups are Galois over some p-adic field?

I'll upgrade my comment to an answer. Any finite Galois extension of $\mathbb{Q}_l$ of degree coprime to $l$ is tamely ramified. In particular, its Galois group is an extension of two cyclic groups. …
Daniel Loughran's user avatar
3 votes
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Understanding the structure of unitary groups

As you already seem to know, unitary groups are classified by separable quadratic field extensions (in fact one should really work with separable quadratic algebras, i.e. also $F\times F$, correspondi …
Daniel Loughran's user avatar