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1d
comment Every neighborhood $N_r(x)$ in $\mathbb{R}^n$ is connected
The rational numbers, $\Bbb Q$, are a perfectly good metric space, and the only connected subsets are the singletons.
1d
answered Expressing a function in terms of compositions of three functions.
2d
answered Simultaneous equation with fractional solutions.
2d
comment division algebras in Weil's Basic Number Theory
I don’t have the book, and I imagine that many people in a position to answer your second question don’t, either. Please give more background. As to your question about finite-dimensionality, it’s probable that he simply omitted that hypothesis.
Jul
4
revised Is it true that group of real numbers under addition isomorphic to group of complex numbers under addition
expansion
Jul
4
comment How many integral ideals $\mathfrak{a}$ are there with the given norm $\mathfrak{N}(\mathfrak{a})=n$?
And therefore I deleted my original comment.
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!