Articles of abstract algebra

Prove that $a=b$, where $a$ and $b$ are elements of the integral domain $D$

This question already has an answer here: $a^m=b^m$ and $a^n=b^n$ imply $a=b$ 6 answers

Question on group homomorphisms involving the standard Z-basis

Let $(e_1, …, e_n)$ be the standard $\Bbb Z$-basis of $\Bbb Z^n$. Let $x_1,…, x_n$ be elements of an abelian group $G$. Prove that there exists a homomorphism $f$ $:$ $\Bbb Z^n$ $\rightarrow$ $G$ such that $f$$(e_i)$ $=$ $x_i$ for all $i$. Can anyone please help me out here?

Who named “Quotient groups”?

Who decided to call quotient groups quotient groups, and why did they choose that name? A lot of identities such as $$\frac{G/A}{B/A}\cong \frac{G}{B}$$ suggest that whoever invented the notation understood these things a lot better than I do… Edit: I’m also interested in the notation; I was assuming that the notation and terminology went together, […]

In a Dedekind domain every ideal is either principal or generated by two elements.

Prove that in a Dedekind domain every ideal is either principal or generated by two elements. Help me some hints. Thanks a lot!

Coefficient of $n$th cyclotomic polynomial equals $-\mu(n)$

For a polynomial $f=X^n+a_1X^{n-1}+\ldots+a_n \in \mathbb{Q}[X]$ we define $\varphi(f):=a_1 \in \mathbb{Q}$. Now I want to show that for the $n$th cyclotomic polynomial $\Phi_n$ it holds that $$\varphi(\Phi_n)=-\mu(n)$$ where $\mu(n)$ is the Möbius function. What I know is that $\displaystyle\Phi_n=\prod_{d|n} (X^{\frac{n}{d}}-1)^{\mu(d)}$.

Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$

How do I show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$? Someone says I should use the rational root test, but I don’t exactly know how that applies. Thanks for any input.

Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.

I’ve tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.

For which $d<0$ is $\mathbb Z$ an Euclidean Domain?

This question already has an answer here: Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD? 2 answers

consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n

consider the group $G=\mathbb Q/\mathbb Z$. Let $n$ be a positive integer. Then is there a cyclic subgroup of order $n$? not necessarily. yes, a unique one. yes, but not necessarily a unique one. never

Proving a basic property of polynomial rings

I am learning ring theory in the Dummit & Foote’s Abstract Algebra, and I am doing all the exercises to get as much experience as possible… but some of them just get me stuck for hours! Like this one : Assume $R$ is a commutative ring with identity. Prove that $p(x)=a_n x^n + a_{n-1} x^{n-1} […]