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10h
comment Split the integral?
If you allow A and f to vary, and the integral is over a fixed interval, then g is in a sense determined. Look at the Riesz representation theorem.
Aug
31
answered Identification between wedge product and its dual
Aug
29
comment Arakelov class group
Look at the treatment of ideles in chapter IV.
Aug
29
comment Arakelov class group
It is in his book Basic Number Theory.
Aug
29
comment Arakelov class group
I do not have that book in front of me. Is that the group of idele classes of norm 1? If so, the usual proof of its compactness is to prove that $O_K^{\times}$ is finitely generated and the class group of $K$ is finite, and then prove one direction of that equivalence. Weil found a method of more directly proving the compactness, but it is harder than the usual proof.
Aug
25
comment Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?
@6005, yes, indeed it is! When I read that related question before I thought the q there was prime, which it need not be.
Aug
25
comment Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?
Another source for a proof is Lemma 6.3.10 of Henri Cohen's "Number Theory, Volume 1: Tools and Diophantine Equations."
Aug
25
comment Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?
related question: math.stackexchange.com/questions/321765/…
Aug
25
comment Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?
Yes. This is proved in Shimura's book "Introduction to the Arithmetic Theory of Automorphic Functions." I don't have a copy in front of me to find the page.
Aug
24
comment What is $\varphi(0)$?
There is no purpose in defining $\varphi(0)$, or certainly no important purpose. I have never seen any reason to need a definition of $\varphi(0)$. This is quite different from the case of deciding whether 1 should or should not be a prime number.
Aug
23
revised Constant rank theorem: intuition?
added 1 character in body
Aug
23
comment What is $\varphi(0)$?
But Pete Clark raised an important point: defining $\varphi(0)$ is completely useless. There is never any reason you need such a definition. This is different from the case of defining $n!$ at $n = 0$.
Aug
23
comment Express roots in polynomials of equation $x^3+x^2-2x-1=0$
@Ali, the answer by Jyrki addresses the connection between this problem and Galois theory.
Aug
23
comment Express roots in polynomials of equation $x^3+x^2-2x-1=0$
@robjohn, yes, I had a sign error in that root in my comment.
Aug
22
comment Express roots in polynomials of equation $x^3+x^2-2x-1=0$
The other two roots are $a^2-2$ and $-a^2-a-1$. How much Galois theory do you know?
Aug
19
comment Proving that the set of separable elements over a field is a field itself.
@Lubin, I would try to prove this without focusing on $a+b$ or $ab$, and prove that if $a$ and $b$ are separable over $k$ then every element of $k(a,b)$ is separable over $k$. Writing out all the details can be tedious, or at least not intuitive, rather than genuinely hard.
Aug
19
comment Is this a correct way of thinking about diffeomorphic manifolds?
The correspondence you describe between functions is "external" only if you refuse to think of functions on a space as being important objects in their own right. Not all the activity of interest is about the spaces; often it is the functions on the spaces that one really cares about. In this light, consider the following posts: math.stackexchange.com/questions/226736/… and mathoverflow.net/questions/21090/smooth-gelfand-duality.
Aug
18
comment Splitting field of $x^4+3$
The last sentence is incorrect. Try $s = -a^2$ for a nonzero integer $a$.
Aug
18
comment Splitting field of $x^4+3$
But is there truly a substantial difference in the ideas for the example you know and the one you ask about?
Aug
18
comment Proving that the set of separable elements over a field is a field itself.
This is a hard result in general. Does your book not prove the result later on? What book is it?