Articles of abstract algebra

Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$

Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$. This is my proof to show that a ring is commutative if it has the property that ab = ca implies b = c. We need to show that if x, y ∈ […]

Abelianization of general linear group?

I am asking purely out of interest: What the abelianization of general linear group $GL(n,\mathbb{R})$?

Is there a homomorphism from a full product of finite cyclic groups onto $\mathbb Z$?

Trying to answer this question, I encountered the following question, the answer to which should be known but it is hard to Google, so I did not find it. Let $G=\prod_{n\in\mathbb N}\mathbb Z_n$ be a full product of finite cyclic groups $\mathbb Z_n=\mathbb Z/n\mathbb Z$. Exists there a homomorphism $f:G\to\mathbb Z$ such that $f((1,1,\dots))=1$? According […]

Non abelian group with normal subgroup

Normal Subgroups are subgroups where all left cosets are right cosets. For abelian groups all subgroups are normal. I want to discuss about a non-abelian group whose subgroups are all normal. Please give an example. Can we give example of a finite non-abelian group with same property ?

Artin's proof of the order of $\mathbb Z/(a+bi)$

As Ben suggested in my earlier question on the subject, I looked at Artin’s proof that $\left|\cdot\right|^2$ is a “size function” which makes $\mathbb Z [i]$ into a Euclidean domain. To quote page 398: We divide the complex number b by a: $b=aw$, where $w=x+yi$ a complex number, not necessarily a Gauss integer. The we […]

On Symmetric Group $S_n$ and Isomorphism

I use Abstract Algebra by Dummit and Foote to study abstract algebra! At page 120, section 2 in chapter 4, there is a great result form my point of view which proves that, for any group $G$ of order $n$, $G$ is isomorphic to some subgroup of $S_n$. My question: Is there any way to […]

Irreducible $f(x) \in F$ of prime degree, $E/F$ finite extension, $p \mid $.

Let $F$ be a field and let $f(x) \in F[x]$ be irreducible of prime degree $p$. Let $E/F$ be a finite extension. Prove: If $f(x)$ is not irreducible in $E[x]$, then $p \mid [E:F]$. (Hint: Consider a field $L$ with $E \subseteq L$ and $L$ as a root of $\alpha$ of $f(x)$.) Proof: Let $F$ […]

The uniqueness of a special maximal ideal factorization

The following problem is from Michael Artin’s Algebra, chapter 12, M.6, unstarred: Let $R$ be a domain, and let $I$ be an ideal that is a product of distinct maximal ideals in two ways, say $I=P_1\dotsb P_r=Q_1\dotsb Q_s$. Prove that the two factorizations are the same, except for the ordering of the terms. Well, following […]

Profinite and p-adic interpolation of Fibonacci numbers

On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here) For each profinite integer $s$, one can in a natural way define the $s$th Fibonacci number $F_s$, which is itself a profinite integer. Namely, given $s$, one can choose a sequence of positive integers $n_1, n_2, n_3,\dots$ that […]

Existence of Homomorphism so that Diagram Commutes

Let $G, H$ be groups and let $\phi: G \to H, \psi: G \to K$ be homomorphism such that $\ker \phi \subseteq \ker \psi.$ Prove that there exists a homomorphism $\theta: H\to K$ such that $\theta\circ\phi = \psi.$ Obviously one can find a homomorphism $\theta$ on $\phi(G)$ which satisfies the properties, but I don’t see […]