Articles of abstract algebra

Finitely generated integral domain and finitely generated $k$-algebra.

Let $k$ be a field and $R$ a finitely generated domain over $k$. Then there are $y_1, \ldots, y_n$ such that $$R=k[y_1, \ldots, y_n]/P,$$ where $P$ is some prime ideal of $k[y_1, \ldots, y_n]$. My question is the following: is $R$ a finitely generated $k$-algebra? It seems that every finitely generated $k$-algebra has the form […]

Euler Totient Issues

I was skimming again through Dummit & Foote’s Abstract Algebra and I came across this exercise: Prove that for any given positive integer $N$ there exist only finitely many integers $n$ with $\varphi(n)=N$, where $\varphi$ denotes Euler $\varphi$-function. Conclude in particular that $\varphi(n)$ tends to infinity as $n$ tends to infinity. I don’t doubt that […]

Does $K/E$ and $E/F$ being normal mean $K/F$ is normal?

Let $F\subset E \subset K$ be fields. Suppose that $K/E$ and $E/F$ are normal. Is $K/F$ also normal? I feel that this statement is not true in general but I cannot find a counter-example. Any thought?

Computing $\operatorname{Tor}_1^R(R/I,R/J)$

I am trying to convince myself that for any ring $R$ (commutative, so I don’t have to bother with left-or-right modules) and ideals $I$, $J$ we have $\operatorname{Tor}_1^R(R/I,R/J)=I\cap J/IJ$. I have found several solutions on the internet using the projective resolution $0\to I\to R\to R/I\to 0$. I am having a hard time understanding why $I$ […]

Proof of $G=N\rtimes H$ iff $G=NH$ and $N\cap H=\{1\}$

I have the following definition of a semi-direct product: Let $G$ be a group. Suppose $N\triangleleft G$ and $H<G$ such that every element of $G$ can be uniquely written $g=nh$. Then $G$ is the semi-direct product of $N$ and $H$. I have to prove the following lemma: Let $N \triangleleft G$ and $H<G$. Then $G=N\rtimes […]

Isomorphism involving tensor products of homomorphism groups

This post is related to the following question: A counterexample to the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$. I have been trying to isolate what hypothesis can be eliminated in these questions surrounding the tensor products of homomorphism groups and I was wondering if the following questions was formulated in such a way as […]

Locally compact nonarchimedian fields

Is it true that if $F$ is a locally compact topological field with a proper nonarchimedean absolute value $A$, then $F$ is totally disconnected? I am aware of the classifications of local fields, but I can’t think of a way to prove this directly.

A doubt with a part of a certain proof.

Well, in the proof of the following lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at, They say that since $H\cap(C+K)= K$, we have $H\cap C=\{e\}$. But how do they have that this part $H\cap(C+K)= K$? Can someone […]

In which of the finite groups, the inverse of Lagrange's Theorem is not correct?

This is a multiple choice for finite groups. For which one of the following groups, the converse of Lagrange’s Theorem is not generally satisfied? I know the converse is true for cyclic groups. 1) All abelian groups 2) All groups of order 8 3) The group $S_4$ 4) All groups of order 12 Thank you […]

Showing that the diagonal of $G \times G$ is maximal, where $G$ is simple

I have been trying to prove the following: Let $G$ be simple, and write $\Gamma=G \times G$. Let $D \le \Gamma$ be the diagonal subgroup, which consists of all elements of the form $(x,x)$, where $x \in G$. Show that $D$ is a maximal subgroup of $\Gamma$. As a hint I am given: Write $\Gamma=A […]