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 Apr 20 comment The irreducibility of $X^q - 3$ in the field extension $\mathbb{Q}(\zeta_q, 2^{1/q})$ For using Kummer Theory, you’d need to show that $3$ remains square-free in $L$, or some such thing that looks to me very difficult. Apr 20 comment The irreducibility of $X^q - 3$ in the field extension $\mathbb{Q}(\zeta_q, 2^{1/q})$ Certainly if $q$ is relatively prime to $3$. Then, the field $\Bbb Q(\zeta_q,2^{1/q})$ is unramified above $3$, but you have $q$ of ramification at $3$ when you adjoin $3^{1/q}$. To get insight, you might try $q=9$ and $q=27$. Apr 19 comment Proving that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers) Not merely fine, but economical. Apr 19 comment An operation on a whole is equal to an operation on each part? Excellent response. I like to say that IN MATHEMATICS, NOTHING IS TRUE. Except when we have a proof that it’s true. OP’s teacher should have explained why $\sqrt a/\sqrt b=\sqrt{a/b}$. If(s)he didn’t, (s)he abdicated all responsibility for teaching. Apr 19 comment How to find inverse of a relation if the inverse isn't a function? As a function defined on $\Bbb Z^{>0}$, this looks perfectly one-to-one and onto to me. There should not be any problem at all. Have you written out the values for inputs from (say) $1$ through $12$? Apr 19 comment How to find inverse of a relation if the inverse isn't a function? There’s always an inverse relation, as subset of, in this case, $\Bbb Z\times\Bbb Z^+$. Apr 19 answered Splitting field in R[X] Apr 18 comment $4$th root of unity: 5-adic The next step is to calculate $7^5\pmod{125}$. These computations do get tedious, more so as you go on, unless you have access to some kind of symbolic calculation package. Apr 18 comment Stronger form of Hensel's lemma? Just as a comment, I hope that you recognize that this is the Newton-Raphson method applied to the problem of finding a root in $\Bbb Z_p$ of $f$. And as to your Idea, I find it’s much easier, once you’re used to it, to use the additive valuation $v_p$ than the absolute value. Apr 17 comment Proving $f(x)$ is not a square in $k[x]$ The polynomial rings are isomorphic, and the corresponding fraction fields likewise. Apr 17 answered If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$? Apr 17 comment Proving $f(x)$ is not a square in $k[x]$ I’m not sure I understand. Aren’t the fields $k(X)$ and $k(x)$ isomorphic? Apr 17 comment How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$? Alternatively, the third vector $y+z$ is the sum of the other two, if the field is $\Bbb F_2$. Apr 16 comment If $L=K(a_1,a_2,..,a_n)$ can we find an irreducible polynomial in $K[x]$ s.t $p(a_i)=0$? You mean one irreducible polynomial good for all the $a_i$’s? If that’s what you mean, then as @quid says, Of Course Not. Apr 16 comment Find the degree measure of an arc of a circle Do you know the theorem that the if $A$, $B$, and $C$ are three points on the circumference of a circle, then the measure of angle $ACB$ is half the measure of the arc $AB$? Apr 16 comment Find the degree measure of an arc of a circle You need to know the theorem that $[m(BC)-m(AD)]/2=30^\circ$. I think that with that, you can answer the question. Apr 16 comment Subgroups of $\mathbb{Z}_p^n$ Yes, if you’re only interested in the isomorphism type. But if you want to count, say, how many $k$-dimensional subspaces of $(\Bbb Z/(p))^n$ there are, that’s another issue. Apr 16 comment What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$? Yeah, my position is that it’s not the least bit intuitive, it’s just provable. That’s good enough. Apr 16 comment What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$? Except that you can convert the familiar (?) formulas of spherical trigonometry to corresponding formulas for hyperbolic trigonometry by making a judicious replacement of sine, cosine, and tangent (of great-circle arcs on the sphere) by sinh, cosh, and tanh for segments of geodesics in the hyperbolic plane. Apr 16 comment Sum of cube roots of a quadratic @AndréNicolas. thanks for that insight: I was about to say that it was hard to know what was meant by “$\sqrt[3]a\,$”