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12h
comment Matrix $B \in M_n(S)$, for $S$ an $R$-algebra, with $R$-independent entries, $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent?
@user26857 Linearly independent over $R$.
14h
asked Matrix $B \in M_n(S)$, for $S$ an $R$-algebra, with $R$-independent entries, $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent?
Jun
25
comment Is a Galois group of a number field faithfully represented by its action on the set prime ideals of the ring of integers?
Do you know if it is possible to get a faithful representation even if we consider just the primes that ramify?
Jun
24
accepted Is a Galois group of a number field faithfully represented by its action on the set prime ideals of the ring of integers?
Jun
24
revised Is a Galois group of a number field faithfully represented by its action on the set prime ideals of the ring of integers?
added 13 characters in body
Jun
24
asked Is a Galois group of a number field faithfully represented by its action on the set prime ideals of the ring of integers?
Jun
24
accepted Every finite abelian extension of Q contains a totally real subfield of index 2?
Jun
23
revised Every finite abelian extension of Q contains a totally real subfield of index 2?
edited title
Jun
23
comment Are there weak versions of the axiom of choice equivalent to weak versions of Zorn's lemma and similar principles?
@AsafKaragila Okay, thanks. So can I consistently refuse to believe both 1) the axiom of choice holds AND 2) there are countable families of countable sets with uncountable union?
Jun
23
comment Are there weak versions of the axiom of choice equivalent to weak versions of Zorn's lemma and similar principles?
@AsafKaragila Can you give an example where it is impossible to construct a bijection without having countable choice? (Which I guess would amount to a countable set of sets with the property that their union being countable implies countable choice?) I am a little more at peace with this result after thinking longer on what it means to be countable ... but it still is weird. (I accept your answer that "everything infinite is counterintuitive.")
Jun
23
answered Quotient module and submodule.
Jun
23
revised Every finite abelian extension of Q contains a totally real subfield of index 2?
[Edit removed during grace period]
Jun
23
comment Are there weak versions of the axiom of choice equivalent to weak versions of Zorn's lemma and similar principles?
How can the countable union of countable sets possibly be uncountable? I can see that you use choice to pick the sequence of bijections, but it breaks my brain to think of this result as possibly not being true. Please help me reconcile this! (You logicians and your draw dropping one liners...!)
Jun
21
comment Good examples of (families of) tamely ramified extensions?
I had been taking the following as my definition: If $p$ is a prime in $Z$, then the extension is tamely ramified above $(p)$ if the ramification index of any prime above $(p)$ is relatively prime to $p$. Is this not a standard definition? It is essentially the same as the first one in the link you provided. @LuisGomezSanchez
Jun
21
revised Good examples of (families of) tamely ramified extensions?
deleted 4 characters in body
Jun
21
asked Good examples of (families of) tamely ramified extensions?
Jun
18
comment Group action on finite set of integers: is it necessarily a permutation group?
An action of $G$ on $\{1, \ldots, n\}$ is the same thing as a homomorphism from $G \to S_n = Aut \{1, \ldots, n\}$. If the action is faithful, meaning that the only group element that fixes all the letters is the identity, then this gives an embedding of $G$ into $S_n$. In particular, if you let a finite group $G$ act on its elements, this gives an embedding of G into $S_n$, for $n = |G|$.
Jun
16
accepted Relationship between Hyperboloid model of hyperbolic space and disc model / confused by a picture.
Jun
16
asked Relationship between Hyperboloid model of hyperbolic space and disc model / confused by a picture.
Jun
13
asked Hopf algebra associated to $GL_V$ for $V$ infinite rank?