Articles of abstract algebra

Galois Group of $\sqrt{1+\sqrt{2}}$

I’m studying algebra, and came upon the following problem. Let $E = \mathbb{Q}(a)$, where $a = \sqrt{1 + \sqrt2}$. Find the irreducible polynomial of $a$, and determine the degree of $E$ over $\mathbb{Q}$. Identity the Galois group of $E/ \mathbb{Q}$, and find how many subfields of $E$ there are. I can see that the irreducible […]

Suppose that half of the elements of G have order 2 and the other half form a subgroup H of order n. Prove that H is an abelian subgroup of G.

Let $n>1$ be a positive integer. Let $G$ be a group of order $2n$. Suppose that half of the elements of G have order 2 and the other half forma a subgroup H of oder n. Prove that H is an abelian subgroup of G. I can only deduce from the index of H that […]

Show that, if $ a + bi$ is prime in $\mathbb{Z} $, then $a – bi$ is prime in $\mathbb{Z}$

Show that, if $ a + bi$ is prime in $\mathbb{Z} [i]$, then $a – bi $ is prime in $\mathbb{Z}[i]$ Since $\mathbb{Z}[i]$ is $ED$, then if $a+bi$ is irreducible then $a+bi$ is prime. But now how I can relate $a+bi$ with $a-bi$? Thanks for help.

Solve $f^n(x)=4^nx+\frac{4^n-1}{3}$ for $n$.

Solve $y=4^nx+\frac{4^n-1}{3}$ for $n$, where $n\in\mathbb{N_{\geq0}},$ $y\in2\mathbb{N}_{\geq0}+1$ and $x\in2\mathbb{N}_{\geq0}+1\setminus(4\mathbb{N}_{\geq0}+1\setminus8\mathbb{N}_{\geq0}+1)$. In clearer language the process is to start with some odd integer and subtract one and divide by $4$ repeatedly until you hit some number which would take you out of the odd integers if you continued further. The question asks you to find for any […]

Isn't $x^2+1 $ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z?$

Isn’t $ x^2+1$ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z[x]$? $ x^2+1$ cannot be broken down further non trivially in $\mathbb Z[x]$. hence, it’s irreducible in $\mathbb Z[x]$. Hence, shouldn’t $\mathbb Z[x]/\langle x^2+1 \rangle$ be a field and hence, $ x^2+1$ a maximal ideal in […]

How to build a subgroup $H\leq S_4$ having order $8$?

Can anyone explain me what would be the procedure for building a subgroup $H\leq S_4$ of order $8$? I started obviously as $H=\{id$. Then I added two disjoint $2$-cycles $(1\ 2), (3\ 4)$ for they commute and they are equal to their inverse, that is, $$H=\{id, (1\ 2), (3\ 4), $$ then I added the […]

Transcendence degree of $K$

Let $K$ be field. How do I proof that transcendence degree of $K[X_1,X_2,\ldots,X_n]$ is $n$? The set $\{X_1,X_2,\ldots,X_n\}$ is algebraically independent over $K$. So, I have to show that every subset of size greater than $n$ is algebraically dependent.

Is $x^6 + 3x^3 -2$ irreducible over $\mathbb Q$?

It seems that it is irreducible over $\mathbb Q$, I tried to apply Eisenstein’s criterion to $f(x+a)$ for some $a$, but it didn’t work. Also, I found a root $\alpha = \sqrt[3]{\frac{\sqrt{17}-3}{2}}$ and tried to prove that the degree of $\alpha$ over $\mathbb Q$ is 6, but I stuck in computation. Is there any approach […]

Fertile fields for roots of unity

Is there a field $K$, an odd prime $p$, and a positive integer $n$, such that $K[ζ] = K[ζ^p]$ where $ζ = ζ_{p^n}$ is a primitive $p^n$th root of unity not contained in $K$? In other words, can a base field $K$ be chosen such that adjoining a “small” root of unity automatically adjoins some […]

If number of conjugates of $x\in H$ in $G$ is $n$ then number of conjugates of $x$ in $H$ is $n$ or $n/2$

Let $G$ a finite group and $H$ subgroup of index $2$. Let $x\in H$ so that the number of conjugates of $x$ in $G$ is $n$. Show that the the number of conjugates of $x$ in $H$ is $n$ or $n/2$.