Given an exact sequence of vector spaces $$\cdots\longrightarrow V_{i-1}\longrightarrow V_{i}\longrightarrow V_{i+1}\longrightarrow\cdots$$ I want to show that it is the same as having a collection of short ones such that $$0\longrightarrow K_i \longrightarrow V_i \longrightarrow K_{i+1}\longrightarrow0$$ So to start I want to show exactness at an arbitrary $V_i$, so I space them suggestively: $$\begin{array}{c} 0&\rightarrow &K_{i-1}&\rightarrow […]

Let $A$ be a commutative ring, $I$ an ideal of $A$ and $M$ an module over $A$. Is it true that $\operatorname{Ann}(M/IM) = \operatorname{Ann}(M) + I$? One inclusion is certainly true, but I don’t know about the other one. If the above statement does not hold in general, does it perhaps for finitely generated modules?

I’m asked to show that $G=GL_2(\Bbb Z/2\Bbb Z)$ is isomorphic to $S_3$. I have few ideas but I don’t manage to put them all together in order to obtain a satisying answer. I first tried using Cayley’s theorem ($G$ is isomorphic to a subgroup of $S_6$), and I also noticed that $\operatorname{Card}(G)=\operatorname{Card}(S3)=6$ & that they’re […]

$\zeta_{15}$ is $15$th primitive $n$th root of unity. Question: Find the structure of the group $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$ I know that if $p$ is prime then $G=Gal(\mathbb{Q}(\zeta_{p})/\mathbb{Q})=\mathbb{Z_{p}}^*$ but when $p$ is not prime, I am not show how to solve this if not prime. Does my required group have any relation to some cyclic group $\mathbb{Z_n}$, i.e. […]

maybe that’s an idiot question. Given a finite field $k$ and some irreducible polynomial $f(x) \in k[x]$, then $k_f \cong k[x]/(f(x))$? I know that it’s true if $k$ is the prime field and I think that the statement is not true for general finite fields, however I could not find any counter example. Thanks in […]

Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?

Undergraduates learn about algebraic objects with one operation, namely groups, and we learn about algebraic objects with two “compatible” operations, namely rings and fields. It seems natural to then look at algebraic objects with three or more operations that are compatible, but we don’t learn about them. I asked one of my professors why this […]

I’m beginning to learn some Grothendieck’s algebraic geometry and I have a doubt about a property of commutative algebra. For a comm. ring $A$ and an ideal $I$ of $A$, does the one-to-one correspondence between ideals of the quotient $A/I$ and ideals of $A$ containing $I$ extends to a correspondence of prime ideals ? My […]

Let $n\in \Bbb N$ be fixed and $m\in \Bbb N$ be the least number such that there exists a group of order $m$ in which all groups of order $n$ can be (isomorphically) embedded. Can we deduce $n!=m$?

It is well-known that for any group $G$ there is an exact sequence $0 \rightarrow \text{Inn}(G) \rightarrow \text{Aut}(G) \rightarrow \text{Out}(G) \rightarrow 0$. Does this sequence always split, i.e. is it always true that $\text{Aut}(G)$ is a semidirect product of $\text{Inn}(G)$ and $\text{Out}(G)$? I suspect not, but have not yet come across a counterexample. Thanks in […]

Intereting Posts

Homomorphism from $\mathbb{Q}/ \mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}$
The Impossible puzzle (“Now I know your product”)
On equivalent definitions of Ext
Coupon collector problem for collecting set k times.
Integrate $\int_0^1 {\frac {x^a-x^b} {\ln x} dx}$
Average value of a complex valued function on a circle.
Reading, Writing, and Proving Math: Cartesian Product
When does the boundary have measure zero?
Proving: If $|A\times B| = |A\times C|$, then $|B|=|C|$.
(Olympiad) Minimum number of pairs of friends.
If $(e_1,…,e_n)$ is an orthonormal basis, why does $\operatorname{trace}(T) =\langle Te_1,e_1\rangle +\cdots+\langle Te_n,e_n\rangle $?
Why would some elementary number theory notes exclude 0|0?
Area of an ellipse. (Calculus)
Order of Cyclic Subgroups
Measurable functions and compositions