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Apr
10
answered Can a Subset be considered an Element for Field Axioms
Apr
9
comment prove n divides $[\mathbb{F}[\alpha]:\mathbb{Q}]$
Don’t you have a chain $\Bbb Q\subset\Bbb Q(\alpha)\subset\Bbb F(\alpha)$? What is $[\Bbb Q(\alpha) : \Bbb Q]$?
Apr
9
answered The elements in the composite field FK
Apr
8
answered proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $
Apr
8
comment Find the Laurent Expansion of $f(z)$
Definitely the quickest way to do it, and the best, in my opinion.
Apr
8
comment Irreducibility in $k((t))[y]$
Yes, to both comments. The upshot is that in most cases, $f(y)-t$ will be reducible.
Apr
8
comment What method was used here to expand $\ln(z)$?
It’s missing the even terms because it’s an odd function: replace $z$ by $-z$ and your fraction turns upside down, so its logarithm changes sign as well. PS: the Grammar Pedant says that the word in this case is not “jive” but “jibe”.
Apr
8
comment Irreducibility in $k((t))[y]$
I don’t believe that the characteristic is a significant part of this story.
Apr
8
answered Irreducibility in $k((t))[y]$
Apr
8
comment Algebra: Field Theory Question
Yes, that is indeed true.
Apr
8
comment How to show that the ring of integers of $\mathbb Q(cos\frac{2\pi}{m})$ is $\mathbb Z(2cos\frac{2\pi}{m})$?
I’m sure you’ve noticed that $2\cos(2\pi/m)=\zeta_m+\zeta_m^{-1}$, where $\zeta_m$ is the primitive $m$-th root of unity $\exp(2i\pi/m)$. I think that if you know how to show that $\Bbb Z[\zeta_m]$ is the integers of $\Bbb Q(\zeta_m)$, then the proof of your proposition should be similar.
Apr
8
comment Problem on a square group
made a slight edit. Hope you don’t mind.
Apr
8
revised Problem on a square group
stuck in a pair of parentheses for clarity
Apr
8
comment Proof for multiplication of two power series
One way is to take $1+u+u^2+u^3+u^4$ and square it, but look only at the terms of degree $\le4$. You should see then what’s happening, and be able to construct a proof, probably with some elements of mathematical induction in it.
Apr
8
comment What does this notation mean? $x \mapsto f(x)$
It says that the if value of the input of your function $f$ is $x$, then the corresponding output is $f(x)$.
Apr
7
comment Prove that the following polynomial is not dividable over Q
Do you know the Rational Root Theorem?
Apr
7
answered A question related to Galois theory
Apr
6
comment How are all the roots of unity of cyclotomic extension are of this form?
Puzzling further over your question, I wonder whether you’re simply asking whether in the field $K_n$generated by the $n$-th roots of unity, these are the only roots of unity in $K_n$. Of course this isn’t true, in case $n$ is odd. If $n$ is even, however, it’s true.
Apr
6
comment How are all the roots of unity of cyclotomic extension are of this form?
Umm, perhaps I might point out that if $n=2$, then $\zeta_n=-1$. I earlier gave you an up-vote, and now I’m not sure I should have.
Apr
6
comment How are all the roots of unity of cyclotomic extension are of this form?
I was going to offer another answer to this question, and then realized that you had not specified what $\zeta_n$ was. Did you mean it to be a primitive $n$-th root of unity?