Reputation
17,505
Next privilege 20,000 Rep.
Access 'trusted user' tools
Badges
1 21 45
Newest
 Generalist
Impact
~185k people reached

Apr
1
comment Construct the rule in fields extensions
Well, if you got your $A$ and your $B$, then $B(\theta)$ is the reciprocal of $g(\theta)$, isn’t it?
Apr
1
comment Find all complex numbers such that $z^4 = 8\bar{z}$
Have you tried anything at all? If this is a homework problem, you should say so.
Apr
1
comment Construct the rule in fields extensions
You know that $f(x)$ and $g(x)=7+2x+x^2$ are relatively prime polynomials, and that there therefore are polynomials $A(x)$ and $B(x)$ such that $Af+Bg=1$ Substitute $x\mapsto\theta$ and get $B(\theta)g(\theta)=1$. You should have seen how to find an $A$ and a $B$ that do the trick for you.
Apr
1
comment $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$?
The idea that the previous two commenters were leading you towards is this: if $f(X)$ is the minimal polynomial for your given irrationality $\alpha$, you want to show that your field is already the splitting field of $f$. This means showing that the other roots of $f$ are already in $\Bbb Q(\alpha)$.
Mar
31
comment equivalent hensel lemma in the case of the rings Z/p*l Z
I’m sorry, I can’t understand your question. In particular, I‘m not sure what “$p*l$” is. Is that the product of two primes? If so, your quotient ring is not of characteristic zero.
Mar
31
answered Factorization of polynomial in a complete field
Mar
31
answered Show that that the ring $\mathbb{Z}_n$ need not have the property $a^2 \equiv a$ implies $a \equiv 0 \pmod{n}$ or $a \equiv 1 \pmod{n}$
Mar
31
comment Why use eccentricity instead of ratio of major and minor axes of an ellipse?
Depending on your intended use of the ellipse, some other measure of thinness than eccentricity might be called for. Mathematically, the nice thing about $e$ is that it’s a uniform parameter that makes sense for parabolas and hyperbolas as well as ellipses.
Mar
30
comment Which of these two factorizations of $5$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{29})}$ is more valid?
Good. It’s really hard doing arithmetic in a real quadratic number field, ’cause the ominipresence of units thoroughly obscures the situation.
Mar
30
comment Are there number systems or rings in which not every number is a product of primes?
This is very nice. We (almost) pure algebraists tend to forget and ignore the vast universe of useful examples from analysis.
Mar
30
comment For which $p$ is a number a square in $\mathbb{Q}_p$?
Yes, but Hensel is what you want.
Mar
30
comment Topics in elliptic curves over finite fields
I guess you’re an undergraduate? When I was around that level, I found amazing the fact that if $E$ is defined over $k=\Bbb F_p$, then the number of points of $E(k)$ determines completely how many points there are of $E$ over the extension of degree $n$, no matter what $n$.
Mar
30
comment Galois group of $x^5-x+1$ over $\mathbb{F}_7$
I guess your error is to think that there’s something leaving your $\alpha_1$ and $\beta_1$ fixed, but moving something else. Maybe this is an implicit assumption that the extension $\Bbb F_7(\alpha_1,\beta_1)$ is not already Galois. But your $\gamma$ is already in the field $\Bbb F_7(\alpha_1,\beta_1)$.
Mar
30
answered Galois group of $x^5-x+1$ over $\mathbb{F}_7$
Mar
30
comment Galois group of $x^5-x+1$ over $\mathbb{F}_7$
No, the two subfields are (linearly) disjoint, so there’s a nontrivial autom of the quadratic field leaving the cubic field elementwise fixed, and vice versa. Let me put it into an answer.
Mar
30
comment Are there number systems or rings in which not every number is a product of primes?
In general, $p=\mathfrak p_1^{e_1}\cdots\mathfrak p_m^{e_m}$, by the factorization of ideals in the integers of the extension field. The $e_i$ are the ramification indices. Take one of the primes in the factorization, say $\mathfrak p_i$. Then, if the ring of integers in the extension is $R$, you have a field $R/\mathfrak p_i$, finite, of degree $f_i$ over the prime field $\Bbb F_p$. That $f_i$ is the residue field extension degree for that upstairs prime.
Mar
30
answered Are there number systems or rings in which not every number is a product of primes?
Mar
29
comment Galois group of $x^5-x+1$ over $\mathbb{F}_7$
Your two separate Galois groups are cyclic of order $2$ and $3$, respectively. Certainly $C_2\times C_3$, cyclic of order $6$, embeds nicely into $S_5$.
Mar
29
comment Conformal Map from Upper Half Plane to Region
Because $z\mapsto z^4$ multiplies the vertex angle (at $0$) by $4$.
Mar
29
comment Inverting the elements of a basis of a finite field extension
Very pleasing counterexample.