Articles of abstract algebra

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 […]

Finding the elements of a finite field?

How do I find the elements of a finite field. For example I wanted to find the elements of $\mathbb{F}_4$ and I constructed an addition table for this: $$\begin{align}+&&0&&1&&\omega && \omega +1\\0&&0&&1&&\omega&&\omega+1\\1&&1&&0&&1&&0\\\omega&&\omega&&1&&0&&\omega+1\\\omega+1&&\omega+1&&0&&\omega+1&&0\end{align}$$ But I am unsure how to make the multiplication table here, not the point however. How would I find the elements for $\mathbb […]

Show that in a field always $0\ne1$

Suppose that $F$ is a field and prove that $0\ne1$ According to the definition of a field I know that the zero element is different from the one element, but is there a scientific proof for that?

Ideals generated by two elements

The question is this: Let $f:\mathbb C[x,y]\rightarrow\mathbb C[t]$ be the homomorphism that sends $x\mapsto t+1$ and $y\mapsto t^3-1.$ Determine the kernel $K$ of $f$, and prove that every ideal $I$ of $\mathbb C[x,y]$ that contains $K$ can be generated by two elements. Solution: I have shown that $((x-1)^3-y-1)=K.$ However, I am having trouble showing the […]

Matrix linear algebra generators

Linear algebra and special-linear group experts please help: It is known that in principle one can generate this $C$ matrix form the $A$ and $B$ matrix below. Here $$ C=\begin{pmatrix} 0& -1& 0\\ 1& 0& 0\\ 0& 0& 1 \end{pmatrix} $$ from: $$ A=\begin{pmatrix} 0& 0& 1\\ 1& 0& 0\\ 0& 1& 0 \end{pmatrix}, \text{ […]