# Tagged Questions

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

198 views

### Irreducible but not prime element

I am looking for a ring element which is irreducible but not prime. So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\in R$. This is irreducible because in any ...
28 views

91 views

79 views

### For which category (if any) are Lie algebras the algebras of a monad?

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...
46 views

### How to prove a given subset is a subgroup [on hold]

If $A, B$ are additive subgroups of a ring $(R,+,\cdot)$, then prove that the set $AB=\{r\in R:r=\sum_{i=1}^n a_i b_i \textrm{ for }a_i\in A,b_i\in B\}$ is an additive subgroup of $R$
33 views

21 views

### How can I prove that the the number of elements of order $k$ in $\mathbb{Z}_n$ is φ(k)? [on hold]

How can I prove that the number of elements of order $k$ in $\mathbb{Z}_n$ is ϕ(k) where $k$ is number that divides n ?
60 views

### If $G$ is a non-abelian group of order 10, prove that $G$ has five elements of order 2.

I'm trying to prove this statement: If $G$ is a non-abelian group of order $10$, prove that $G$ has five elements of order $2$. I know that if $a\in G$ such that $a\neq e$, then as a ...
618 views

### Problems from the Kourovka Notebook that undergraduate students can fully appreciate

The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that are, ...
112 views

### Infinitely many primes $\equiv 3 \mod 4$

Question 1 Is the following proof of the infinitude of primes $\equiv 1\mod 4$ okay? Consider a prime divisor $p\mid (n!)^2+1$. Then $(n!)^2\equiv -1 \mod p$, hence $n!$ has multiplicative order $4$ ...
### Is an algebraic field extension $k \subseteq K$ normal if $\mathrm{Aut}_{K}(\bar{k})$ is a normal subgroup of $\mathrm{Aut}_{k}(\bar{k})$?
Over a perfect field $k$ it is well known that an algebraic field extension $k \subseteq K$ is normal if and only if $\mathrm{Aut}_{K}(\bar{k})$ is a normal subgroup of $\mathrm{Aut}_{k}(\bar{k})$, as ...