# Sanchez

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I really appreciate it when you take time to answer my questions. Thanks!

 Dec4 comment What is the value of $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G$ are in the same conjugacy class? Or more simply, note that it's the sum of the trace of $g$ acting on $Hom(\rho_i, \rho_i)$ if $\rho_i$ accounts for all irrep of $G$. But $\oplus Hom(\rho_i, \rho_i) = \mathbb{C}[G]$, so you are looking for the trace for $g$ acting on the group algebra. This is $|G|$ if $g=1$ and 0 otherwise. Dec4 comment Dimension of the tangent space of an algebraic set Your $k$ is not $\mathbb{R}$ (which isn't even algebraically closed!). So your (a) and (c) have problems. Dec4 comment Is there an elementary proof that there are infinitely many primes p satisfying the following: It doesn't feel like Chebotarev is necessary - Dirichlet's theorem on arithmetic progression should already be enough, if you are willing to get your hands dirty to show directly for instance 2 is a QR but not a fourth root mod p iff p satisfies some congruence conditions, which would of course follows from biquadratic reciprocity. Dec3 comment Arithmetic in p-adic You don't need Hensel's lemma to know that there is no solution in $\mathbb{Z}_5$: The natural homomorphism $\mathbb{Z}_5 \to \mathbb{F}_5$ shows that if there is a squareroot of 2 in $\mathbb{Z}_5$, then there is one in $\mathbb{F}_5$, and you know that the latter is false. It's the converse where Hensel's came in, and as Luis said, you will probably just prove Hensel's on the way without realizing it. I do think this is the only way of doing it. Dec3 comment Is there an elementary proof that there are infinitely many primes p satisfying the following: It seems that biquadratic reciprocity has to come in somewhere - is there any reason you think that an elementary proof exists? Dec3 comment Amount of Background Needed for Number Theory Research Depending on your interest, if you go into number theory, you will along the way pick up some of representation theory, harmonic analysis, algebraic geometry, ergodic theory, mathematical physics etc, and it's likely that you can also do research in these areas. Maybe I'm wrong, but from your comments I get the impression that you think if you devote all your time to number theory then you won't have enough background for other areas. This is simply true, not only for number theory, but for most branches of Mathematics; many different areas of math are intertwined closely. Dec3 comment Arithmetic in p-adic The general procedure of these things is that, first you want to solve your equation in $\mathbb{F}_p$. In your case, it's clear that $x^2 = 2$ has a solution mod 7 (say 3), but not mod 5. Once you solved it in mod $p$, Hensel's lemma allows you to lift the solution uniquely to mod $p^2$, $p^3$ and so forth, which allows you to put them all together and solve it in $\mathbb{Z}_p$. This can be generalized to cubic root, or more general polynomial equations, as long as the condition of Hensel's lemma is satisfied. Dec3 comment What is so special about frobenious endomorphisms in elliptic curves? I guess in the context of elliptic curves, a great use of Frobenius is that its fixed points (say, you are over $\mathbb{F}_p$) are exactly the $\mathbb{F}_p$ points of the elliptic curves. One can then use insights from topology, in particular Lefschetz fixed point theorem, to study the number of $\mathbb{F}_p$ points of the elliptic curve. (And of course, more generally $\mathbb{F}_{p^n}$ points) This leads to the Grothendieck-Lefschetz trace formula. Dec3 comment Integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{x\sin{x}}{1+\cos^4{x}}dx$ 1. You are right, sorry. 2. I'm not familiar with polylogarithms, but as in comments above 0912 claimed that it has a solution in terms of hypergeometric series, what you said is possibly right then. Dec3 comment Integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{x\sin{x}}{1+\cos^4{x}}dx$ It's not linear functions of $x$, it's a function of $\cos x$ that involves squares and square root. See wolframalpha.com/input/?i=integrate+1%2F%281%2Bx^4%29 Dec3 comment Integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{x\sin{x}}{1+\cos^4{x}}dx$ And how do you integrate $F(x)$? Dec3 comment What is the value of $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G$ are in the same conjugacy class? Your teacher is right, I misread the indices. In that case, use your other question to get that the character table is an orthogonal matrix (after you suitably normalize each term). If the other question was column orthogonality, this one would be row orthogonality. Dec2 comment What is the value of $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G$ are in the same conjugacy class? Yes, from your other question. Dec1 comment What do we know about the class group of cyclotomic fields over $\mathbb{Q}$? Thanks! This should really be a new question, but here it goes anyway: What would non-abelian reciprocity law tell us in this case (characterizing the set of completely splitting primes in $H$)? It's not congruence conditions on $p$, but what kind of condition would it be? Dec1 comment What do we know about the class group of cyclotomic fields over $\mathbb{Q}$? Thanks! I know this (actually that's how I came up with the question). I meant to characterize the set of primes in $\mathbb{Q}$, that splits completely in $H$ (the Hilbert class field of $\mathbb{Q}(\zeta_q)$ in this case, not necessarily abelian like you said) in a way similar to class field theory. I know that congruence conditions won't be possible here, but what can I say in general? Dec1 comment How prove this matrix inequality $\lambda_{n-1}\le\frac{n}{n-1}\min{\{a_{jj}:1\le j\le n\}}$ @user1551, thanks! Dec1 comment How prove this matrix inequality $\lambda_{n-1}\le\frac{n}{n-1}\min{\{a_{jj}:1\le j\le n\}}$ Why is $x^*Ax \ge \lambda_{n-1} \|x \|^2$? $\lambda_{n-1}$ is not the smallest eigenvalue. Nov30 comment Question about Radon-Nikodym derivative If your measure is positive, then any $\mu$ would dominate any $\nu$ given your condition, since the only $\mu$-null set is the empty set. For the Radon-Nikodym derivative, write $\nu = \int_A f d \mu$, and evaluate it at each singleton set $\{x\}$ - you will be able to write down what $f$ is then at each $x$. Nov30 comment Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$ Do you know Jensen's inequality? Note that $f(x) = \log x$ is concave. The right upper bound should be $\frac{\log 2}{2}$. Nov30 comment What is the value of $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G$ are in the same conjugacy class? But it's the same as $\sum \chi_i (g) \chi_i(g^{-1})$, since $h^{-1}$ and $g^{-1}$ are in the same conjugacy class. Can you evaluate the former sum?