# Questions tagged [abstract-algebra]

For questions about groups, rings, fields, vector spaces, modules and other algebraic objects. Associate with related tags like group-theory, ring-theory, modules, etc. to clarify which topic of abstract algebra is most related to your question and help other users when searching.

57,551 questions
38 views

### Set Theory, Infinite chain of subsets [on hold]

Given an infinite chain of subsets $S_1\subseteq S_2 \subseteq S_3...$. Consider a subset $N\subset\bigcup\limits_{i=1}^{\infty} S_{i}$, can we say that $N\subseteq S_i$ for some $i$. If so, how do we ...
40 views

### Size of conjugacy class in subgroup compared to size of conjugacy class in group

Given: $\bullet$ A finite group $G$, an index 2 subgroup $H$, an element $a \in H$ $\bullet$ $[a]_H$ and $[a]_G$ are the conjugacy class in $H$ of $a$ and the conjugacy class in $G$ of $a$, ...
14 views

### Prove: A nonempty subset B of a ring A is closed with respect to addition and negatives iff B is closed with respect to subtraction [duplicate]

Can I say something like, ∀ x,y ∈ B, x + (-y) ∈ B ⇒ x - y ∈ B by definition of subtraction?? I'm lost
22 views

### Group as direct sum of cyclic groups

What are necessary conditions for a cyclic group $G$ to be a direct sum of cyclic groups? I saw somewhere that $G$ must be a non $p$-group. But I couldn't prove it. Thank you for your hints/help
50 views

47 views

### Product of $\prod_{i=0}^{\infty}(1+q^{2^{i}})$ [duplicate]

I know that answer to $\prod_{i=0}^{\infty}(1+q^{2^{i}})$ = $(1+q)(1+q^{2})(1+q^{4})...$ = $\frac{1}{1-q}$. And i need to prove this equality using generating functions. Any hints?
27 views

### A question regarding finding the minimal polynomial associated with a field extension . [duplicate]

Say we have the field extension $\Bbb Q(w,\sqrt[3]{5})$ over $\Bbb Q$, where w is the primitive cubed root of unity. I know that the minimum polynomial of $\sqrt[3]{5}$ is $x^3-5$. I want to figure ...
47 views

### Generators for $\mathbb{Z_n}$

I would like to show that $K$ is a generator for $\mathbb{Z}_n$ $\iff$ $\gcd(K,n)=1$ and $1 \leq K <n$. My Attempt: Assume $\gcd(K,n)=1$ and $1 \leq K <n$. That means $K \in \mathbb{Z}_n$ and ...
28 views

36 views

109 views

### Finding degree of a finite field extension

Let $x=\sqrt{2}+\sqrt{3}+\ldots+\sqrt{n}, n\geq 2$. I want to show that $[\mathbb{Q}(x):\mathbb{Q}]=2^{\phi(n)}$, where $\phi$ is Euler's totient function. I know that if $p_1,\ldots,p_n$ are ...