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KConrad

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Name KConrad
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Jun
14
 comment Does there exist an order in a number field of deg>1 with a map to F_p for all p?
Related topic: if a monic irred. polynomial in ${\mathbf Z}[x]$ has a root mod $p$ for all but finitely many primes $p$ then the poly. has degree 1. This is proved by the "$H = G$" conjugation result you ask about. Its connection to your question is that if $\alpha$ is a root of that irred. polynomial and $R = {\mathbf Z}[\alpha]$, then finding a ring homomorphism $R \rightarrow {\mathbf F}_p$ amounts to finding a root of the polynomial mod $p$. If your question had "all but finitely many $p$" then the order would be in $\mathbf Q$, and ${\mathbf Z}[1/N]$ maps to ${\mathbf F}_p$ if $(p,N)=1$.
Jun
14
 comment Does there exist an order in a number field of deg>1 with a map to F_p for all p?
That every conjugacy class is a Frobenius conjugacy class for a positive density of primes is generally attributed to Chebotarev rather than Dirichlet.
Jun
14
 comment Does every equivalence class of Hecke characters contain a distinguished element?
The integer $n$ is not specified anywhere. Was a condition left out?
Jun
11
 comment Why is it a good idea to study a ring by studying its modules?
Mariano: a quotient of two ideals is not a ring, just as an ideal in a ring (other than the whole ring) is not a subring. At least not in commutative algebra.
Jun
10
 revised Longest coinciding pair of integer sequences known
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Jun
8
 comment The equation x^2+2=y^3 admits a unique solution in positive integer
The first step is to switch the roles of $x$ and $y$, since $y^2 = x^3 - 2$ is the usual way it is written nowadays. Then read about Mordell's equation, which can be found in many references on Diophantine equations or elliptic curves. Your question is not a research-level question, so it's more suitable for math.stackexchange if you need further assistance.
Jun
8
 awarded ● Enlightened
Jun
8
 awarded ● Nice Answer
Jun
7
 revised Are there Carlitz analogues of quadratic residues and reciprocity?
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Jun
7
 accepted Are there Carlitz analogues of quadratic residues and reciprocity?
Jun
7
 revised Are there Carlitz analogues of quadratic residues and reciprocity?
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Jun
7
 comment Difficulty understanding Russian- Paper in Galois Theory
What paper did Shafarevich publish in 1956? There's no paper from that year on MathSciNet, except for a translation into German of a paper from 1954. Have you considered looking in Shafarevich's Collected Papers, which are translations of his work into English?
Jun
7
 revised Are there Carlitz analogues of quadratic residues and reciprocity?
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Jun
7
 revised Are there Carlitz analogues of quadratic residues and reciprocity?
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Jun
6
 revised Are there Carlitz analogues of quadratic residues and reciprocity?
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Jun
6
 revised Are there Carlitz analogues of quadratic residues and reciprocity?
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Jun
6
 answered Are there Carlitz analogues of quadratic residues and reciprocity?
Jun
4
 comment Can repunits be perfect cubes?
For $y^2 = x^3 + k$ with $k \not= 0$: in ${\mathbf Z}_2$ use $(x,k+1)$ s.t. $x^3 = (k+1)^2 - k$, in ${\mathbf Z}_3$ use $(1-k,y)$ s.t. $y^2 = (1-k)^3+k$, in ${\mathbf Z}_5$ use $(x,y)$ s.t. $x^3 + k \equiv 1 \bmod 5$. In ${\mathbf Z}_7$, use $(0,y)$ if $k \equiv 1, 2, 4 \bmod 7$, $(1,y)$ if $k \equiv 0, 3 \bmod 7$, $(-1,y)$ if $k \equiv 5 \bmod 7$, and $(x,0)$ if $k \equiv 6 \bmod 7$. For $p \geq 11$, let $N_p$ be number of mod $p$ solns, so $N_p = p + S_p$ with $S_p=0$ if $p|k$ and $|S_p| \leq 2\sqrt{p}$ by Hasse if $(p,k)=1$. Then $N_p \geq 4$; lift mod $p$ soln w/ $y \not\equiv 0$ mod $p$.
Jun
4
 comment Can repunits be perfect cubes?
so there's a mod $p$ solution $(x,y)$ with $y \not\equiv 0 \bmod p$, and such a mod $p$ solution can be lifted $p$-adically.
Jun
4
 comment Can repunits be perfect cubes?
@Noam: You're right that there's no congruence argument: $y^2 = x^3 + 7$ is solvable in the $p$-adic integers for all primes $p$. First let's treat $p \leq 7$. If $p = 2$ use $(x,0)$ s.t. $x^3 = -7$. If $p = 3$ use $(0,y)$ s.t. $y^2 = 7$. If $p = 5$ use $(-1,y)$ s.t. $y^2 = 6$. If $p = 7$ use $(1,y)$ s.t. $y^2 = 8$. If $p > 7$, let $N_p$ be the number of $(x,y) \bmod p$ s.t. $y^2 \equiv x^3 + 7 \bmod p$. Then $N_p = p + \sum (\frac{x^3+7}{p})$, w/ sum being over $x \bmod p$. Call the sum $S_p$, so $|S_p| \leq 2\sqrt{p}$ by Weil's bound. Then $N_p \geq p - 2\sqrt{p} > 3$ (since $p>9$) [contd]
Jun
1
 comment Representation quaternions as matrices
The Artin-Wedderburn theorem doesn't claim there is a representation as 2 x 2 matrices. Take Hamilton's quaternions $A(-1,-1)$ with $F = {\mathbf R}$. The usual representation with matrices uses 4 x 4 real matrices or 2 x 2 complex matrices.
May
29
 revised What is interesting/useful about big Witt Vectors?
Fixed year of Crelle from 1936 to 1937
May
28
 comment Linear Algebra Texts?
There is a note after the preface of the English translation saying that Gelfand asked for the appendices not to be translated. I looked at them online (they're on perturbation theory) and couldn't figure out why the appendices would not be something to translate. Do you know if there was some awkward homage to Stalin in the 1950 edition? The appendices I looked at are from a more recent edition.
May
25
 awarded ● Nice Answer
May
24
 awarded ● Good Answer
May
24
 awarded ● Guru
May
24
 revised Relations between automorphisms of field of rational functions and Mobius Transfomation
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May
24
 revised Relations between automorphisms of field of rational functions and Mobius Transfomation
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May
24
 answered Spinoffs of analytic number theory
May
21
 comment How much of character theory can be done without Schur’s lemma or the Artin-Wedderburn theorem?
Frobenius defined characters and proved theorems about them before there was a definition of representation of a group. See ams.org/notices/199803/lam.pdf, esp. (6.4) and (7.4). He used the group determinant in place of representations (e.g., a character is irreducible when it corresponds to an irreducible factor of the group determinant).
May
21
 revised sum of squares in ring of integers
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May
21
 revised sum of squares in ring of integers
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May
21
 revised sum of squares in ring of integers
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May
14
 awarded ● Nice Answer
May
2
 comment Why is it hard to prove that the Euler Mascheroni constant is irrational?
For a general number field $K$, Ihara introduced in 2006 an Euler-Kronecker constant $\gamma_K$, which is best defined from the Laurent expansion at $s=1$ not of $\zeta_K(s)$, but rather of $\zeta_K'(s)/\zeta_K(s)$: $\zeta_K'(s)/\zeta_K(s) = -1/(s-1) + \gamma_K + O(s-1)$. In terms of the Laurent expansion $\zeta_K(s) = R/(s-1) + c + O(s-1)$, we have $\gamma_K = c/R$. For $K = {\mathbf Q}$, $R = 1$ and therefore $c = \gamma_{\mathbf Q}$. But for general $K$ the number $R$ is not 1 and the constant term of $\zeta_K(s)$ at $s = 1$ is the "wrong" object of interest.
May
2
 comment Why is it hard to prove that the Euler Mascheroni constant is irrational?
Because philosophers are not likely to provide a key idea?
Apr
29
 revised Origins of names of algebraic structures
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Apr
20
 comment Upper bound on order of finite subgroups of GL_n(Z_p)?
Concerning Qiaochu's question, the bound certainly has to involve something about $R$, such as its rank over ${\mathbf Z}_p$, because otherwise you could let $R$ be the integers in the degree $d$ unramified extension of ${\mathbf Q}_p$ and use the group of matrices ${\rm diag}(a,1,\dots,1)$ where $a$ runs over the group of $p^d-1$-th roots of unity in $R$. That is a group of order $p^d-1$ inside ${\rm GL}_n(R)$ and its order gets arbitrarily large as $d \rightarrow \infty$. So probably the question should have the upper bound expressed as $C(n,p,r)$, where $r ={\mathbf Z}_p$-rank of $R$.
Apr
17
 comment how to visualize the class number of an imaginary quadratic field?
Misha's comment extends to all number fields, e.g., see the introduction to math.fsu.edu/~petersen/Petersen-countingcusps.pdf.
Apr
16
 accepted D K Faddeev’s construction of quaternionic fields
Apr
16
 revised D K Faddeev’s construction of quaternionic fields
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Apr
16
 comment D K Faddeev’s construction of quaternionic fields
One more reference: in a survey on Galois theory at the Steklov Institute (books.google.com/…) Faddeev writes at the end of section 5 that the algebraic part of his construction of quaternionic Galois extensions extends to any base field outside characteristic 2 with enough quadratic extensions.
Apr
16
 comment D K Faddeev’s construction of quaternionic fields
Hmm, I noticed that the beginning of the article I cited is very similar to an article on Faddeev for his 80th birthday (already written in English): mathsoc.spb.ru/pantheon/faddeev/UMN-89e.html. The only difference in the part I wrote above is that the one written for his 80th birthday includes subgroups of $S_3$ in the list before subgroups of $S_4$.
Apr
16
 comment D K Faddeev’s construction of quaternionic fields
I too am intrigued by what his geometric construction could be, since I am teaching Galois theory at the moment and such a construction of an extension of ${\mathbf Q}$ with quaternionic Galois group would be great to show the class (better than presenting explicit field generators out of nowhere). If someone has easy access to the papers, could they indicate what the construction in them is?
Apr
16
 answered D K Faddeev’s construction of quaternionic fields
Apr
6
 comment Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
What is the "singular value map" in terms of the singular value decomposition? Does it associate to a matrix the diagonal matrix in is singular value decomposition?
Apr
6
 comment Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
The second statement should also have the summation index $i$ running explicitly from 1 to $n$.
Apr
6
 comment Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
Since you are summing over the $n$th symmetric group while you have a list of $m$, rather than $n$, matrices, could you make the highlighted inequalities clearer by writing explicitly what the range of summation is on the right side ($i$ runs from what to what)?
Apr
1
 comment Domain of the wedge product in Little Spivak
Although the exterior powers are arguably better thought of as a quotient space, and that way of treating them is essential for exterior powers of modules over a general commutative ring, my impression (as an outsider) is that in differential geometry and physics it is still common to regard the exterior powers as subspaces of antisymmetric tensors.
Apr
1
 comment On mentioning recommenders' names in cover letter for postdoctoral applications
If someone applied for a job through Mathjobs and did not include a cover letter at all, it would look unprofessional, just as much as applying for a job ouside academia and not including a cover letter. Even if it is a minor part of the application (certainly compared to the recommendation letters and research statement), it should be written. Most of the time it'll be the same letter at all schools.