Articles of abstract algebra

Show that $ a,b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $

Inspired by this, I was wondering if there is a simple logical argument to Show that $ a,b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $ Note that the original link is using a computational method, where as I am looking for a simple logical argument. I tried (unjutifiably) to argue that if some […]

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ which sends $3$ mod $4$ to $-1$.

Working out a concrete example of tensor product

From this entry in Wikipedia: The tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$. It is denoted $V\otimes_K W$, or $V\otimes W$ when the underlying field $K$ is understood. If $V$ has a basis $e_1,\cdots,e_m$ and $W$ has a basis $f_1,\cdots,f_n$, then the tensor […]

How to prove that the converse of Lagrange's theorem is not true?

I consider the Lagrange theorem. Let $G$ be a finite group and let $H \subseteq G$ be a subgroup, then the order of $H$ divides the order of $G$. I am interesting with the proof of this theorem. The proof is as follows Let $C= \{a_1 H, a_2 H,\ldots,a_t H\}$ be a collection of all […]

The (Jacobson) radical of modules over commutative rings

Let $M$ be a module over a commutative ring $R$. Let $\Omega$ be the set of all maximal ideals of $R$. Prove that $\operatorname{Rad}(M)=\bigcap_{\mathfrak m\in \Omega}\mathfrak mM$, where $\operatorname{Rad}(M)$ is the intersection of maximal submodules of $M$. (This is exercise 15.5, p. 174, from Anderson and Fuller, Rings and Categories of Modules.) Thanks for the […]

Direct product of two normal subgroups

Let $A$ and $B$ be normal subgroups of a group $G$ such that $A \cap B = \langle e \rangle$ and $AB = G$. Prove that $A \times B \cong G$ Attempted proof: Define $f : A \times B \rightarrow G$ by $f(a,b) = ab$. From a proof of another exercise, the hypothesis of this […]

Quotient of polynomials, PID but not Euclidean domain?

While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the Euclidean domain page of Wikipedia that $$\mathbb{R}[X,Y]/(X^2+Y^2+1)$$ is such a ring. After a good deal of searching, I have not been able to find any other (online) reference to this ring. Can anyone confirm […]

Order of matrices in $GL_2(\mathbb{Z})$

Let $A\in GL_2\left(\mathbb{Z}\right)$, the group of invertible matrices with integer coefficients, and denote by $\omega(A)$ the order of $A$. How we prove that $$\left\{\omega(A);A\in GL_2\left(\mathbb{Z}\right)\right\}=\{1,2,3,4,6,\infty\}.$$

Properties of set $\mathrm {orb} (x)$

Properties of set $\mathrm {orb} (x)$: ${\displaystyle \bigcup_{x\in X}\mathrm{orb}(x)=X}$; $\mathrm{orb}(x)\cap\mathrm{orb}(y)=\emptyset$ for all $x,y\in X, x\neq y$ How to prove it? Please help. Appedix: Let $\phi: G \times X \longrightarrow X$ – action of the group G on the non-empty set $X$. The set $\mathrm {orb} (x) = \{ \phi (g,x) \in X: g \in G […]

For what $n$ is $U_n$ cyclic?

When can we say a multiplicative group of integers modulo $n$, i.e., $U_n$ is cyclic? $$U_n=\{a \in\mathbb Z_n \mid \gcd(a,n)=1 \}$$ I searched the internet but did not get a clear idea.