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 $R[\mathbb{Z}]$ […]

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 […]

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 […]

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 […]

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 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 $a,b \in K$ […]

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 […]

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 $pq$ […]

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 […]

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 $\Rightarrow E \supset L$ […]

Intereting Posts

Definite integrals with interesting results
Integral Representation of the Zeta Function: $\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$
Finding a specific improper integral on a solution path to a 2 dimensional system of ODEs
Can an odd perfect number be divisible by $5313$?
Where to start learning Linear Algebra?
$\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint.
Injective Holomorphic Functions that are not Conformal?
How can I sum the infinite series $\frac{1}{5} – \frac{1\cdot4}{5\cdot10} + \frac{1\cdot4\cdot7}{5\cdot10\cdot15} – \cdots\qquad$
Can this approach to showing no positive integer solutions to $p^n = x^3 + y^3$ be generalized?
Show that $\lim_{n\to\infty}b_n=\frac{\sqrt{b^2-a^2}}{\arccos\frac{a}{b}}$
Non-linear function on $\mathbb{R}^2$ preserving the origin and maps lines onto lines?
How to solve $A\tan\theta-B\sin\theta=1$
Prove that $|\log(1 + x^2) – \log(1 + y^2)| \le |x-y|$
Most ambiguous and inconsistent phrases and notations in maths
Abstract algebra book recommendations for beginners.