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comment How can I have a copy of this old paper by Frobenius?
Are you aware that the papers of Frobenius were gathered into his Collected Works in the late 1960s? It won't help with a translation, of course, but it means you can find all his papers bound together in these volumes in many university libraries.
Oct
19
comment Is the notation ${}^t g$ for the transpose of a linear transformation intended to be suggestive?
Frank, it's because the superscript is appearing on the left side, which looks wrong at first. At least this is the reasoning I made up when I first saw it. I never discussed it with anyone.
Oct
19
comment Is the notation ${}^t g$ for the transpose of a linear transformation intended to be suggestive?
I always regarded the placement of it to the left as a reminder that the transpose reverses the order of multiplication. Differential geometers write coordinates as $x^i$, so I never thought that it would be confused with an exponent when it's in the upper right (since the reader ought to know what the context is).
Oct
18
comment Number Theory over $\mathbb{F}_q [t]$, why is it important/interesting?
One way number theory in the function field case ($\mathbf F_q(t)$ and its finite extensions) can be applied back to more classical problems is in strong estimates on classical exponential sums. The best estimates are usually obtained from analogues of the Riemann hypothesis for zeta-functions or $L$-functions in function fields. By the way, in your question, the analogue of $\mathbf R$ in the function field case is not the rational function field $\mathbf F_q(t)$ but Laurent series fields such as $\mathbf F_q((t))$ and $\mathbf F_q((1/t))$.
Oct
18
revised Number Theory over $\mathbb{F}_q [t]$, why is it important/interesting?
deleted 4 characters in body; edited title
Oct
18
comment Number Theory over $\mathbb{F}_q [t]$, why is it important/interesting?
A possible duplicate: mathoverflow.net/questions/1367/…
Oct
18
answered Counterexamples in Algebra?
Oct
15
comment What is the ring $A_{\Gamma}$ in the Cohen-Lenstra Heuristics?
If $\Gamma$ is cyclic of order $n$ then their ring $\mathbf Q[\Gamma]/\sum_{g\in \Gamma} g$ is basically $\mathbf Q[X]/(1+X+...+X^{n-1})$, which is not a field unless $n$ is prime. In general this ring is a product of fields, namely the fields $\mathbf Q[X]/(\Phi_d(X))$ where $d$ runs over the factors of $n$ greater than $1$.
Oct
12
comment Are the first ten zeros of this Dedekind zeta function non-simple?
A similar phenomenon happens for the zeta-function of any number field $K$ that is Galois over $\mathbf Q$ such that ${\rm Gal}(K/\mathbf Q)$ is non-abelian: $\zeta_K(s) = \prod_{\rho} L(s,\rho)^{d_\rho}$, where $\rho$ runs over the different (that is, non-isomorphic) irreducible representations of ${\rm Gal}(K/\mathbf Q)$ and $d_\rho$ is the dimension of $\rho$. When ${\rm Gal}(K/\mathbf Q)$ is non-abelian, some $d_\rho$ is greater than 1, so every zero of $L(s,\rho)$ has multiplicity at least $d_\rho$ as a zero of $\zeta_K(s)$, and thus is a non-simple zero of $\zeta_K(s)$.
Oct
9
comment Rings of algebraic integers as quotients of polynomial rings
For (ii), see mathoverflow.net/questions/21267/…, including the comments.
Oct
9
comment Reference: Bochner Integral`
Have you seen Bochner integration at all and just want to see applications of it in probability theory, or are you asking for a source where you can learn about it with probability theory as the motivating topic?
Oct
7
comment Central limit theorem via maximal entropy
See Remark 4.5 of math.uconn.edu/~kconrad/blurbs/analysis/entropypost.pdf and the reference at the end of it.
Oct
4
comment $p$-adic $n$-th root of unity and $\exp(2\pi i /n)$
That would be a constructive proof that $(\mathbf Z/p\mathbf Z)^\times$ is cyclic for all primes greater than 3, and such an approach is not known. Strictly speaking, you could work in the ring $\mathbf Z[\exp(2\pi i/(p-1)]$ and use prime ideals lying over $p$ as a "model" for $\mathbf Z/p\mathbf Z$, but that is not explicit in the way that you seek.
Oct
4
comment $p$-adic $n$-th root of unity and $\exp(2\pi i /n)$
No. First, the notation $\eta_n$ is not well-defined since there's more than one primitive $n$th root of unity in $\mathbf Z_p^\times$. Let's say you meant to ask if there is some primitive $n$th root of unity in $\mathbf Z_p^\times$ that can be computed from $\exp(2\pi i/n)$. Still the answer is no. Look at it this way: if $n = p-1$ (for $p > 3$) the question asks for a formula for a generator of the $(p-1)$th roots of unity in $\mathbf Z_p^\times$. If you could do that then reducing the formula mod $p$ gives a formula for a generator of $(\mathbf Z/p\mathbf Z)^\times$ for all $p > 3$!
Sep
30
awarded  Explainer
Sep
30
comment Not too classical group characters
Why not tell us what actual applications you have in mind for such characters, to show the students that the concept is useful?
Sep
30
comment Not too classical group characters
For any field $k$ and nonzero $a \in k^\times$, let $\chi \colon\mathbf Z \rightarrow k^\times$ by $\chi(n) = a^n$.
Sep
30
comment Why have mathematicians used differential equations to model nature instead of difference equations
effects of working with finite fields in place of $\mathbf R$, such as there being no meaningful notion of ordering: every number is "negative", in the sense of being the additive inverse of a suitable sum of 1's, many "negative" integers can be perfect squares (in $\mathbf Z/11\mathbf Z$, $-2 = 9$ so $-2$ is a perfect square), and the multiplicative inverse of a "positive" number can be "larger" than it, e.g., in $\mathbf Z/11\mathbf Z$ the inverse of 2 is 6. In reality, does half a meter seem to be larger than 2 meters? This would be completely at odds with how measurements work in reality.
Sep
30
comment Why have mathematicians used differential equations to model nature instead of difference equations
@CraigFeinstein, the idea that there "is" a highest number, which when added to 1 makes 0, makes me wonder: what do you mean by "is"? Do you think the construction of $\mathbf R$ within pure math has a flaw, do you think counting in nature can be modeled more accurately using $\mathbf Z/n\mathbf Z$ for a mysterious large number $n$ (and if so, provide evidence), or something else? I have read one of Zeilberger's rants that we should model reality with $\mathbf Z/p\mathbf Z$ in place of $\mathbf R$ for some huge prime $p$ (to make it a field), but this ignores profound algebraic [contd.]
Sep
23
comment The angular distribution of the $(a,b)$ in $p = a^2+b^2$, and the distribution of the lattices corresponding to prime ideals
The equidistribution theorem for Gaussian primes in the first quadrant is mentioned as an example in Lang's Algebraic Number Theory. See Example 2 in Chapter XV.