Articles of abstract algebra

The Jacobson Radical of a Matrix Algebra

I am trying to solve the following question. Let $A$ be the algebra over $\mathbb{C}$ consisting of matrices of the form \begin{pmatrix} * & * & 0 & 0 \\ * & * & 0 & 0 \\ * & * & * & 0 \\ * & * & * & * \\ \end{pmatrix} […]

What are the conditions for integers $D_1$ and $D_2$ so that $\mathbb{Q} \simeq \mathbb{Q}$ as fields.

What are the conditions for integers $D_1$ and $D_2$ so that $\mathbb{Q}[\sqrt {D_1}] \simeq \mathbb{Q}[\sqrt {D_2}]$ as fields. Here $\mathbb{Q}[\sqrt {D}] := \{a + b \sqrt D \mid a,b \in \mathbb{Q} \}$ Really not sure where to begin with this sort of problem. I was thinking that I should split into cases where the integer […]

Showing the polynomials form a Gröbner basis

Let $A$ be an $m \times n$ real matrix in row echelon form and $I \subset \mathbb{R}[x_1,\dots,x_n]$ is an ideal generated by polynomials $p_i = \sum_{j = 1}^na_{ij}x_j$ with $1 \leq i \leq m$. Then the generators form a Gröbner basis for $I$ w.r.t. some monomial order. So I guess one should try the standard […]

Does $\text{End}_R(I)=R$ always hold when $R$ is an integrally closed domain?

Let $R$ be a commutative ring with identity, and $I\neq 0$ an ideal of $R$, I’m thinking how to calculate $\text{End}_R(I)$. I have proved that when $R$ is a integral domain, $\text{End}_R(I)=\{r\in\text{Frac}(R)\mid rI\subset I\}$. So I’m curious about if $\text{End}_R(I)=R$ always holds when $R$ is also integrally closed. Can anyone help me with this? Thanks […]

How to factor the ideal $(65537)$ in $\mathbb Z$?

This question is related to How to factor ideals in a quadratic number field? In Algebraic Number Theory by W. Stein he makes a remark about the factorization of $65537$ in $\mathbb Z[i]$. I checked this in Sage and the result is different. What is an explanation of this difference?

The intersection of two Sylow p-subgroups has the same order

Let $G$ be a finite group and assume it has more than one Sylow $p$-subgroup. It is known that order of intersection of two Sylow p-subgroups may change depending on the pairs of Sylow p-subgroups. I wonder whether there is a condition which guarantees that intersection of any two Sylow $p$-subgroups has the same order. […]

Ideal in $\mathbb Z$ which is not two-generated

I can prove that the ideal $(4, 2x, x^2)$ in $\mathbb Z[x]$ is not principal. But I failed to prove that this cannot be generated by two elements. It’s really difficult for me. Would you give me a hint.

When is a module over $R$ and $S$ an $R \otimes S$-module?

Suppose $M$ is a module over $R$ and $S$, commutative rings with $1$. Under what conditions is $M$ also a $R \otimes S$-module? Also, a more general question: How to construct a structure of a $R \otimes S$-module? In other words, when one wants to construct a map from $R \otimes S$ to another ring, […]

Module isomorphisms and coordinates modulo $p^n$

Let $p$ be a prime and $n \in \mathbb{N}$ is such that $p^n > 2$. We let $\alpha \in \mathbb{N}$ be such that $0 < \alpha < n$. Let $R := \mathbb{Z}_{p^n}$ denote the ring of integers modulo $p^n$ and let $<p^{\alpha}>$ denote the principal ideal given by the set $\{p^{\alpha}r | r \in R\}$. […]

Show $F(U) = K((x^q -x)^{q-1})$.

Let $K$ be a finite field with $q$ elements. Show that if $U$ is the subgroup of $Aut(K(x)/K)$ which consists of all mappings $\sigma$ of the form $(\sigma \theta)(x) = \theta(ax+b)$ with $a \neq 0$ then $F(U) = K((x^q -x)^{q-1})$. I am not getting any clue to solve the problem. Help Needed. Here $F(U)$ is […]