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Jul
4
answered How many integral ideals $\mathfrak{a}$ are there with the given norm $\mathfrak{N}(\mathfrak{a})=n$?
Jul
4
comment Converges to 0 vs. Diverges to 0; Terminology in p-adic Analysis
By your lights, is it licit to speak of the sequence $\{1+p^n\}$ converging to $1$?
Jul
4
comment Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$
Didn’t you see the technique of Rationalizing the Denominator in high school?
Jul
4
comment Fermat's Theorem and primitive $n$th roots of unity
Just as a matter of experience, I’ve found it rare that the brute-force method usually finds a generator in $\{2,3,5\}$.
Jul
4
comment Is it true that group of real numbers under addition isomorphic to group of complex numbers under addition
A one-to-one and onto map between two bases always extends to the vector spaces, by linearity.
Jul
2
comment extend trace and norm from a number field $K$ to $ K \otimes_{\mathbb Q} \mathbb R $
How do your descriptions of trace and norm differ from the description of $T(a)$ as the trace of the linear map $z\mapsto az$, for $z\in K$, and $N(a)$ as the determinant of this linear map? I’m referring to the regular representation, you see. When you tensor with any $\Bbb Q$-algebra, these definitions are unchanged.
Jul
2
comment converging power series over $p$-adic integers is a UFD
I’d be happy to discuss this via e-mail.
Jul
2
comment converging power series over $p$-adic integers is a UFD
It’s certainly not true that invertibility in $\Bbb Q_p\{x\}$ implies irreducibility in $\Bbb Z_p\{x\}$. Maybe you meant “irreducible in $\Bbb Q_p\{x\}$”? On another topic, I think I may have missed the point somewhat in my answer. Without thinking very much on the topic, it seemed to me last night that the irreducibles in the two rings should be the same, and your desired result should drop out fairly easily from the information that you have in hand.
Jul
2
comment Mixed vs Equal Characteristic Local Rings
Yes, just so. But don’t get the idea that the basis consists of nothing but the monomials!
Jul
2
answered converging power series over $p$-adic integers is a UFD
Jul
2
answered $5^m$, where m is any natural, can be expressed as the sum of two perfect squares?
Jul
1
comment Mixed vs Equal Characteristic Local Rings
Here’s a simple-minded contribution: equal-characteristic rings are power-series rings, you can “see” exactly what’s going on; mixed-characterstic rings are not power-series rings.
Jul
1
comment p-th root does not become a p-th power when adjoined?
My “approach” clearly was wrong, and depended on $k$ being a number field besides, evidently the wrong way to think about the problem, in view of @Slade’s pleasing response.
Jul
1
comment p-th root does not become a p-th power when adjoined?
Looks very nice. Thanks!
Jul
1
comment Evaluating limit (iterated sine function)
Lovers of power series, Unite!
Jul
1
comment For which $d \in \mathbb{Z}$ is $\mathbb{Z}[\sqrt{d}]$ a unique factorization domain?
I would hate to try to compute this by hand, even for $\sqrt{10}$, i.e. $d=40$.
Jul
1
comment p-th root does not become a p-th power when adjoined?
A nice question, as edited. The only approach I can think of involves a tedious division into cases, and I hope someone offers a uniform proof.
Jul
1
comment Evaluating limit (iterated sine function)
Better and more pithy than my suggestion.
Jul
1
answered Evaluating limit (iterated sine function)
Jun
29
comment Non real complex in metric completions of $\mathbb Q$
@LuisGomezSanchez, sorry for the delay, but I was away from wifi. If $\Phi_n$ is the $n$-th cyclotomic polynomial, whose roots are the primitive $p$-th roots of unity, and if $\xi$ is a root of $\Phi_n$, then $\xi$ may be sent into $\Bbb C$ to an $\exp(k\pi i/n)$, but *which $k$*? Similarly, $\xi$ may be sent into $\Bbb Q_p$ if $n|(p-1)$, but in which of the possible $\phi(n)$ ways? There is simply no way to identify which element of a $p$-adic field $\exp(i\pi/n)$ corresponds to.