Reputation
Next privilege 15,000 Rep.
Protect questions
Badges
25 55
Newest
 Nice Answer
Impact
~261k people reached

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?
Aug
18
comment Splitting field of $x^4+3$
Please tell us if you understand any similar problem, e.g., have you learned how to calculate the Galois group of $x^4-2$ over $\mathbf Q$?
Aug
16
comment Find the derivative of $f(x) = \int_{-\infty}^\infty \frac{e^{-xy^2}}{1+y^2}\ dy.$
You forgot to include in the definition of $f(x)$ that $x \geq 0$, or perhaps $x > 0$.
Aug
16
revised Find the derivative of $f(x) = \int_{-\infty}^\infty \frac{e^{-xy^2}}{1+y^2}\ dy.$
edited title