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Mar
16
comment P-adic valuation for ideals
Have you done this for $A=\Bbb Z$?
Mar
14
comment $r=5 \sec(\theta)$ into rectangular
Have you tried graphing a few points? Like for $\theta=\pi/6,\,\pi/4,\,\pi/3$?
Mar
13
comment Intermediate fields for a Cyclotomic Polynomial of order $27$?
I hope that you’ve used the responses of @JyrkiLahtonen to find the Galois group and the intermediate fields. If not, I give an additional hint: for primes $p$ other than $2$, the Galois group of the field of $p^m$-th roots, over $\Bbb Q$, is always cyclic. So there will be precisely one intermediate field of each possible degree.
Mar
12
answered Endomorphisms of the multiplicative formal group law
Mar
12
comment Finding units in cyclotomic fields
But the field of cube roots of unity is not cubic but rather quadratic over $\Bbb Q$. Your basis should have been $\{1,\zeta\}$.
Mar
12
answered Evaluate $(-1)^x$ by hand.
Mar
11
comment Unramified cubic extension of imaginary quadratic fields
it means that your cubic extension $K\supset k$ may not be of the form $K=k(\alpha^{1/3})$, that’s all I was saying. Indeed, any time that $k(\alpha^{1/3})$ is Galois over $k$, you must have $\omega\in K$, and I don’t believe that this is the case in most of the situations we’re talking about, since $3$ is not ramified in $K$.
Mar
11
answered Inverse Limits in Galois Theory
Mar
11
comment Given two arbitrary terms of a geometric series, can this lead to geometric series parameters that cannot be solved for in terms of radicals?
This doesn’t seem to have anything to do with Galois theory. But have you solved this in any specific cases at all? Like, maybe, when $n=m+1\,$?
Mar
10
comment Unramified cubic extension of imaginary quadratic fields
I don’t think that you can expect that your cubic extension will be gotten by adjoining a cube root. You would need $\omega$, a primitive cube root of unity, to be in your smaller field expect that.
Mar
10
comment Finding the minimal polynomial and its conjugates without a matrix
It’s high-school algebra. DON’T use $\omega=\exp(2\pi i/3)$. Simply use the identities $\lambda^3=5$ and $\omega^2+\omega+1=0$. Every time you get an $\omega^2$, replace it by $-\omega-1$, and every time you get a $\lambda^3$, replace it by $5$.
Mar
10
answered Finding the minimal polynomial and its conjugates without a matrix
Mar
10
comment Converse: $f'(x)\ge 0 \implies f$ is monotonically increasing?
Indeed, even for functions that aren’t continuous.
Mar
9
answered How do I find the image of the domain under the map $f(z)=z^2$?
Mar
9
comment Under what conditions is a tower of quadratic extensions a UFD, GCD domain, or just an Integral Domain?
About GCD domains I’ll have to plead ignorance. But in general, when you adjoin a square root, you automatically adjoin all associated integral elements, so that the result will always be in integrally closed domain. For instance, when you adjoin $\sqrt5$ to $\Bbb Z$, you don’t get an integrally closed ring. You have to include half of $1+\sqrt5$ as well.
Mar
9
comment Under what conditions is a tower of quadratic extensions a UFD, GCD domain, or just an Integral Domain?
You’ve put your finger into a most delicious mathematical pie. You need to read, read, read, and take as many courses as you can. As a rough approximation, let me say that the more irrationalities you adjoin to $\Bbb Q$, the less likelly it will be that you get a UFD. Meanwhile, do as much hand computation as you can, and meditate on what you’re finding.
Mar
9
comment Requirement on Norm for units in Cyclotomic Fields
The fact that in algebraic integer is a unit if and only if its norm down to $\Bbb Q$ is $\pm1$ has nothing to do with roots of unity. Each of the exemplary responses below gives a proof valid for the general case.
Mar
9
comment Necessity of algebraic symbolism
Let me say, to concur with the above two comments, that when I was little, I thought the same thing as you. But then when the problems got more complicated, I understood why the general method always worked, but the “common sense” approach almost always got lost in the multitude of details. Teach a general method for handling problems by working on simple ones to persuade the student that the general method is valid. Then let the student discover independently that the general method is superior.
Mar
9
comment Exercise to determine $\mathcal{O}_K$
I would have started with the basis $\{1,(1+\sqrt p)/2\}$ and then passed to the biquadratic ring, till I read the helpful comment of @MooS. You should certainly use what you know about quadratic fields here... And did you look at the case of $3$ and $5$?
Mar
9
comment Unramified cubic extension of imaginary quadratic fields
Aha, but that was exactly why I wasn’t too happy with your remark about class number $3$. Can you “write down” the Hilbert class field in either case? If you can, it should not be too hard to perform the last step in the case of class number $6$.