# Sanchez

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

 Dec15 comment Hartshorne's definition of structure sheaf disjoint union, or coproduct in the category of sets. Dec14 comment If $abc=1$ and $a,b,c$ are positive real numbers, prove that ${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1$. @mathh, if you know Muirhead's inequality, at the final step you just homogenize your inequality (i.e. multiply $(abc)^{2/3}$ to $a+b+c$ part) and use Muirhead. If not, then try to find a clever AM-GM: for example: $a^2b+a^2c \ge 2a^{3/2}$ by AM-GM, then you can use Power mean or other approaches to prove $a^{3/2} + b^{3/2} + c^{3/2} \ge a+b+c$. Dec11 comment Why is this result of the Cauchy-Goursat theorem true? You can also write down the anti-derivative like John said, as long as $n \neq 1$. So if you can see that for polynomials, you should be able to see that for $n \ge 2$ too. As for $n = 1$, interpretation of integrating $dz/z$ as winding number is probably helpful. Dec11 comment Why is this result of the Cauchy-Goursat theorem true? what makes you think that intuitively, integrating polynomial over line is 0? Dec5 comment An exercise in Silverman You are working in $\mathbb{P}^2$ I believe, which means that $[x,y,z] = [tx,ty,tz]$. In particular if $z \neq 0$, you can normalize it to make it 1, i.e. you can always plug in $z = 0$ or $z=1$ in your equations. Dec5 comment Trying to show AM-GM mean inequality It is. It's generally called the weighted AM-GM inequality. Dec5 comment How to find the minimum of $a+b+\sqrt{a^2+b^2}$ Are you forbidding calculus? Dec4 comment How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$ How did you find 4/5? 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.