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Feb
4
comment A nontrivial solution to the quadratic form $x^2 - xy + y^2$ over the finite field $𝔽_p$ with $p ≡ 1 \pmod3$ a prime
You beat me to it by a couple of minutes.
Feb
4
answered A nontrivial solution to the quadratic form $x^2 - xy + y^2$ over the finite field $𝔽_p$ with $p ≡ 1 \pmod3$ a prime
Feb
4
comment Does the determinant give you the index over $\mathcal{O}_k$ as well as over $\mathbb{Z}$?
Yes, I guess that was my point.
Feb
4
comment Upper bound on exact power of wild prime that divides the different
Yes, @JyrkiLahtonen, I think the local case of the following extension of $\Bbb F_p((t))$ also shows this interesting difference from the mixed-characteristic case. Let $\pi$ be a root of $X^p+t^nX+t$. I suppose it shows how there are separable elements arbitrarily close to the inseparable element $t^{1/p}$.
Feb
4
comment An automorphism of the field of $p$-adic numbers
Maybe it’s late to be making a comment here, but I love this argument. I proved the triviality of the automorphism group of $\Bbb Q_p$ for myself by a far more elaborate argument, and this one really appeals to me. But it seems to me that talking about the cubehood of $1+p^2x^3$ should work just as well, and not need any special arguments, since it’s fine at $3$.
Feb
3
comment Does the determinant give you the index over $\mathcal{O}_k$ as well as over $\mathbb{Z}$?
Are you aware of the theorem in algebraic number theory that the index of $Ra$ in $R$ is $|\mathbf N(a)|\,$? And then, locally, one ordinarily defines $v_{\mathfrak P}(a)$ exactly to be $v_p(\mathbf N(a))$. In the latter case, I’m thinking of the complete situation, where one then shows that this $v_{\mathfrak P}$ is a valuation by a Hensel’s-Lemma argument.
Feb
3
comment Does the determinant give you the index over $\mathcal{O}_k$ as well as over $\mathbb{Z}$?
For $1$-by-$1$ matrices $[\alpha]$, the answer is $\mathbf N(\alpha)$. This area is far from my meat, but the problem looks local in nature to me. Can’t you do it at a prime $\mathfrak P|p$ of $K$ and combine all the nonunit results?
Feb
2
answered Modulus Notation Division
Feb
1
comment P-adic norm/valuation of n!
I was preparing an analysis of the convergence properties of the exponential that used only the fact that the closest roots to zero of the logarithmic series $\log(1+x)$ have $v(\rho)=\frac1{p-1}$, but it got so long that I decided it was not a contribution to the discussion. Your answer is the best.
Feb
1
comment P-adic norm/valuation of n!
Are you asking about $x^n/n!\>$? And maybe about the sum of these, the exponential series?
Feb
1
answered Why is the set of units of integer quaternions isomorphic to the quaternion group of order 8?
Jan
31
answered Easy computational example of first cohomology group: Is this how we do it?
Jan
31
comment Easy computational example of first cohomology group: Is this how we do it?
I'm away from a computer now, but will try to get to it in a couple of hours.
Jan
31
comment Easy computational example of first cohomology group: Is this how we do it?
You slipped where you concluded from $\mathcal N^{\Bbb Q(i)}_{\Bbb Q}(z)=1$ that $z$ had to be a unit of $\Bbb Z[i]$. After all, $\frac35+\frac45i$ also has norm $1$. If this doesn’t help you enough, I’ll give a fuller story in an answer.
Jan
31
answered Upper bound on exact power of wild prime that divides the different
Jan
31
comment Complex number ( prove )
This is the argument I would have given, but maybe better than mine, ’cause you put in more intermediate steps than I would have.
Jan
31
revised Put $A\cos(x) + B\sin(x)$ into form : $A\sin(x+ \theta)$
edited body
Jan
31
answered Solving $12x \equiv 20 \pmod{38}$
Jan
29
comment Find roots or splitting field of a polynomial given its Galois group
I left out square root (of discriminant) in the fmla for Mink bound. Corrected
Jan
29
revised Find roots or splitting field of a polynomial given its Galois group
left out asquare root!