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Jul
16
comment Solving $z^z=z$ in Complex Numbers
Problem is that under the condition that $a$ is not positive real and $b$ is not integral, $a^b$ has no unambiguous definition. In particular, when you take $\log$ of both sides to get $z\log z=\log z$, it's not even clear to me that you want the same branch of the logarithm on both sides of the equation.
Jul
12
comment $A_4$ extension of $\mathbb{Q}$ ramified at one prime
To answer this, we need to know how much you already know, what you’ve tried, and also whether it’s a homework problem.
Jul
10
comment Are there invariants of formal group laws other than height?
Just noticed this. I guess you’re aware that for each $p$, there are only finitely many supersingular values of $j$. This is a corner of the topic of the relationship between elliptic curves and formal groups that I’m far from an expert in, but maybe I can help you a little, if you’re still interested in the topic. Don’t hesitate to e-mail me.
Jul
9
comment Galois Group of $x^{4}+7$
’Tweren’t me, but it was a natural misreading. (The extension is indeed of degree eight.)
Jul
9
comment Galois Group of $x^{4}+7$
There are exceptions, but when you adjoin the fourth root of a square-free, you expect the normal closure to be of degree eight.
Jul
9
comment Galois Group of $x^{4}+7$
For a nice set of generators, I would write $\{\sqrt[4]{-7},i\}$, where “$\sqrt[4]{-7}$” just means a once-and-for-all choice of the fourth root. The first guess you made for the splitting field is clearly wrong, since it’s an extension of degree $16$.
Jul
9
comment “Negative” versus “Minus”
Of course, @chharvey, and that’s exactly how I would justify my position if I followed my preferences. But if I’m going to kick against a bad practice of modern teaching, I’ll reserve my kicks for truly damaging pedagogical practices.
Jul
9
comment Is this induction proof of the Fundamental Theorem of Algebra rigorous?
@Prism, my memory is totally unreliable, but this does not look like the proof I saw (about 60 years ago), though I do recall that it involved the Fundamental Theorem on Symmetric Functions.
Jul
8
comment Is this induction proof of the Fundamental Theorem of Algebra rigorous?
There’s a proof using nothing but the fact that a polynomial of odd degree over $\Bbb R$ has at least one real root, but I no longer know it.
Jul
6
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.
Jul
6
answered Expressing a function in terms of compositions of three functions.
Jul
5
answered Simultaneous equation with fractional solutions.
Jul
5
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.