Show that is $F$ is a finite field then, $P_F$ (the set of all polynomial functions on F) is not isomorphic to $F[x]$ It gives the the definition: an element $\phi \in F^F$ is a polynomial function on $F$, if there exists $f(x) \in F[x]$ such that$\phi(a)=f(a), \forall a \in F$ $F[x]$ is infinite in […]

Let $R$ be a finite boolean ring. It’s known that there’s a boolean algebra/ring isomorphism $R\cong \mathcal P(\mathsf{Bool}(R,\mathbb Z_2))$. I’m trying to get a feel for this. The subsets of $\mathsf{Bool}(R,\mathbb Z_2)$ should somehow correspond to elements of $R$. At first I thought of $\mathbb Z_2$ as the usual subobject classifier in the category of […]

Let $k$ be a field and $R$ a finitely generated domain over $k$. Then there are $y_1, \ldots, y_n$ such that $$R=k[y_1, \ldots, y_n]/P,$$ where $P$ is some prime ideal of $k[y_1, \ldots, y_n]$. My question is the following: is $R$ a finitely generated $k$-algebra? It seems that every finitely generated $k$-algebra has the form […]

I was skimming again through Dummit & Foote’s Abstract Algebra and I came across this exercise: Prove that for any given positive integer $N$ there exist only finitely many integers $n$ with $\varphi(n)=N$, where $\varphi$ denotes Euler $\varphi$-function. Conclude in particular that $\varphi(n)$ tends to infinity as $n$ tends to infinity. I don’t doubt that […]

Let $F\subset E \subset K$ be fields. Suppose that $K/E$ and $E/F$ are normal. Is $K/F$ also normal? I feel that this statement is not true in general but I cannot find a counter-example. Any thought?

I am trying to convince myself that for any ring $R$ (commutative, so I don’t have to bother with left-or-right modules) and ideals $I$, $J$ we have $\operatorname{Tor}_1^R(R/I,R/J)=I\cap J/IJ$. I have found several solutions on the internet using the projective resolution $0\to I\to R\to R/I\to 0$. I am having a hard time understanding why $I$ […]

I have the following definition of a semi-direct product: Let $G$ be a group. Suppose $N\triangleleft G$ and $H<G$ such that every element of $G$ can be uniquely written $g=nh$. Then $G$ is the semi-direct product of $N$ and $H$. I have to prove the following lemma: Let $N \triangleleft G$ and $H<G$. Then $G=N\rtimes […]

This post is related to the following question: A counterexample to the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$. I have been trying to isolate what hypothesis can be eliminated in these questions surrounding the tensor products of homomorphism groups and I was wondering if the following questions was formulated in such a way as […]

Is it true that if $F$ is a locally compact topological field with a proper nonarchimedean absolute value $A$, then $F$ is totally disconnected? I am aware of the classifications of local fields, but I can’t think of a way to prove this directly.

Well, in the proof of the following lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at http://torus.math.uiuc.edu/jms/m317/handouts/finabel.pdf, They say that since $H\cap(C+K)= K$, we have $H\cap C=\{e\}$. But how do they have that this part $H\cap(C+K)= K$? Can someone […]

Intereting Posts

Examples when vector $(X,Y)$ is not normal 2D distribution, but X and Y are.
How to reformulate a multiplicative formula (with two primes, perhaps like totient-function)?
Finding singular points and computing dimension of tangent spaces (only for the brave)
Closed form for an infinite sum over Gamma functions?
Why is the Jordan Curve Theorem not “obvious”?
First kind Chebyshev polynomial to Monomials
Automorphisms in $Z_n$
If every $0$ digit in the expansion of $\sqrt{2}$ in base $10$ is replaced with $1$, is the resulting sequence eventually periodic?
homotopic maps from the sphere to the sphere
distribution of categorical product (conjunction) over coproduct (disjunction)
Any group of order four is either cyclic or isomorphic to $V$
Interpolated Fibonacci numbers – real or complex?
What are some interesting sole exceptions or counterexamples?
Expected Value of sum of distinct random integers
Integrating trigonometric function problem $\int \frac{3\sin x+2\cos x}{2\sin x+3\cos x}dx$