Articles of abstract algebra

Giving meaning to $R$ (for example) via the evaluation homomorphism

In our course we introduced the concept of polynomials as a part of a more general construction, namely the monoid ring (or even more general, the monoid algebra) $R[M]$, where $(R,+_R,\cdot_R)$ is a ring and $(M,+_M)$ is a monoid. Then we introduced the notation $R[x]:=R[\mathbb{N}]$. So “$R[x]$” has a precisely defined meaning (Just like $R[\mathbb{Z}]$ […]

Prove that a non-abelian group of order $pq$ ($p<q$) has a nonnormal subgroup of index $q$

So I’ve come up with a proof for the following question, and I’d like to know if it’s correct (as I couldn’t find anything online along the lines of what I did). Question Let $p$ and $q$ be primes with $p<q$. Prove that a non-abelian group of order $pq$ has a nonnormal subgroup of index […]

Infinite Groups with Finitely many Conjugacy Classes

If $G$ is an infinite group with finitely many conjugacy classes, what can be said about $G$? (Should $G$ be simple/ solvable/….?) For $n\geq 2$, does there exists a (infinite) group $G$ with exactly $n$ conjugacy classes, which is periodic also? I couldn’t find any information on these questions. One may provide links also for […]

$(\ker (f))_{\frak{p}}=\ker (f_{\frak{p}})$

Suppose $R$ is a ring and $M,N$ are $R$-modules and $f:M\to N$ is an $R$-linear map. If $\frak{p}$ is a prime ideal of $R$, then we have a $R$-linear map $f_{\frak{p}}:M_{\frak{p}}\to N_{\frak{p}}$ (where $M_{\frak{p}}:=M/\mathfrak{p}M$ and $N_{\frak{p}}:=N/\mathfrak{p}N$) defined by $f_{\frak{p}}([m])=[f(m)]$. I’m trying to prove that $(\ker (f))_{\frak{p}}=\ker (f_{\frak{p}})$. It is obvious to check inclusion $\subset$, but […]

Homogenous polynomial over finite field having only trivial zero

Is there a way to construct homogenous polynomials of small degree over a certain finite field having only trivial zero? For instance, the polynomial $f (a, b, c) = a^3 + b^3 + c^3 – 3abc – 3a^2b – 3b^2c – 3c^2a$ has only a trivial zero over $\mathbb{F}_{13}^3$ (I found this by trial and […]

I don't understand what a Pythagorean closure of $\mathbb{Q}$ is; how are these definitions equivalent?

I have two definitions of the said field. And frankly I don’t see why one is equivalent to the other. It just doesn’t add up. Let’s look at wikipedia’s definition. In algebra, a Pythagorean field is a field in which every sum of two squares is a square. So, say there are $a,b \in K$ […]

Tensor product of Hilbert Spaces

I am following this link under “definitions” I need to see why the suggested inner product on the pre-Hilbert space $H_1$ tensor $H_2$ is well defined. Recall that the fundamental tensors are a spanning set, but not a linearly independent one, hence not free in the category of vector spaces. How do I know that […]

Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center

A nonabelian group $G$ of order $pq$ where $p$ and $q$ are primes has a trivial center My Proof is as follows: Assume we have nonabelian group $G$ of order $pq$ where both $p$ and $q$ are primes. When $G$ has a trivial center it means subgroup $Z(G)=\{e\}$. If a group is of order $pq$ […]

why is a polycyclic group that is residually finite p-group nilpotent?

I am trying to solve an exercise in D. Robinson’s book A Course in the Theory of Groups, which asks me to show that if $G$ is polycyclic and residually finite p-group for infinitely many prime p, then $G$ is nilpotent and finitely generated torsion-free. How do I show the nilpotent part? The hint given […]

Intermediate fields Separable, Algebraic, or Splitting

I just took my algebra final, and I had a few questions about intermediate fields and the properties of their extensions True or False…For $E \supset L \supset F$ $E \supset L$ is algebraic and $L \supset F$ is algebraic $\Rightarrow E \supset F$ is algebraic $E \supset F$ is algebraic $\Rightarrow E \supset L$ […]