Articles of abstract algebra

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} […]

Ring with 10 elements is isomorphic to $\mathbb{Z}/10 \mathbb{Z} $

How do I prove that a finite ring $R$ of order 10 is isomorphic to the ring $\mathbb{Z}/10 \mathbb{Z}$? I know that as a group under addition, $(R,+)$ is isomorphic to the group $(\mathbb{Z}/10 \mathbb{Z}, + )$, but the multiplication is rather mysterious to me.

CW complex such that action induces action of group ring on cellular chain complex.

Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ on the space $\overline{X}$ given by the isomorphism of $\pi$ with $\text{Aut}(\overline{X})$. Let $A$ be an abelian group and let $\mathbb{Z}[\pi]$ act trivially on $A$, $a \cdot […]

Is there a subfield $F$ of $\Bbb R$ such that there is an embedding $F(x) \hookrightarrow F$?

I believe that for any subfield $F$ of $\Bbb R$, with finite degree of transcendence over $\Bbb Q$, the field of rational fractions in one variable $F(x)$ can’t be embedded in $F$. However the situation can be different for those with infinite degree of transcendence over $\Bbb Q$. I know that $F(x)$ always embeds in […]

Appearance of Formal Derivative in Algebra

When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are coprime. My question is, should we be surprised to see the formal derivative here? Is there some way we can make […]

Can we extend the definition of a homomorphism to binary relations?

This is going to be quite a long post. The actual questions will be at the end of it in section “Questions.” INTRODUCTION After receiving an answer to this question about extending the definition of a continuous function to binary relations, I started thinking about doing the same with homomorphisms in abstract algebra. It seems […]

Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $n\ne 4$, is irreducible over $\mathbb Z$

I try to solve this problem. I seems to come close to the end but I can’t get the conclusion. Can someone help me complete my proof. Thanks Show that the polynomial $h(x) = (x-1)(x-2)\cdots(x-n) + 1$ is irreducible over $\mathbb Z$ for all $n \ge 1, n \neq 4$. Suppose $h(x) = f(x) g(x)$, […]

Requesting abstract algebra book recommendations

I’ve taken up self-study of math. (How smart can that be?) I’ve just about finished a course in real analysis which spent a lot of time on metric spaces and some time revisiting calculus. I was thinking of trying abstract algebra. I would appreciate any book recommendations. Thanks in advance. Andrew

Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$

I can’t crack this one. Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that $a,b,c,z \in \mathbb{Z}$).