This question already has an answer here: $a^m=b^m$ and $a^n=b^n$ imply $a=b$ 6 answers

Let $(e_1, …, e_n)$ be the standard $\Bbb Z$-basis of $\Bbb Z^n$. Let $x_1,…, x_n$ be elements of an abelian group $G$. Prove that there exists a homomorphism $f$ $:$ $\Bbb Z^n$ $\rightarrow$ $G$ such that $f$$(e_i)$ $=$ $x_i$ for all $i$. Can anyone please help me out here?

Who decided to call quotient groups quotient groups, and why did they choose that name? A lot of identities such as $$\frac{G/A}{B/A}\cong \frac{G}{B}$$ suggest that whoever invented the notation understood these things a lot better than I do… Edit: I’m also interested in the notation; I was assuming that the notation and terminology went together, […]

Prove that in a Dedekind domain every ideal is either principal or generated by two elements. Help me some hints. Thanks a lot!

For a polynomial $f=X^n+a_1X^{n-1}+\ldots+a_n \in \mathbb{Q}[X]$ we define $\varphi(f):=a_1 \in \mathbb{Q}$. Now I want to show that for the $n$th cyclotomic polynomial $\Phi_n$ it holds that $$\varphi(\Phi_n)=-\mu(n)$$ where $\mu(n)$ is the Möbius function. What I know is that $\displaystyle\Phi_n=\prod_{d|n} (X^{\frac{n}{d}}-1)^{\mu(d)}$.

How do I show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$? Someone says I should use the rational root test, but I don’t exactly know how that applies. Thanks for any input.

I’ve tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.

This question already has an answer here: Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD? 2 answers

consider the group $G=\mathbb Q/\mathbb Z$. Let $n$ be a positive integer. Then is there a cyclic subgroup of order $n$? not necessarily. yes, a unique one. yes, but not necessarily a unique one. never

I am learning ring theory in the Dummit & Foote’s Abstract Algebra, and I am doing all the exercises to get as much experience as possible… but some of them just get me stuck for hours! Like this one : Assume $R$ is a commutative ring with identity. Prove that $p(x)=a_n x^n + a_{n-1} x^{n-1} […]

Intereting Posts

Show that $\dfrac{d}{dt}\left(ml^2\dot\theta\right)+mgl\sin\theta=l\dfrac{d^2\theta}{dl^2}+2\dfrac{d\theta}{dl}+\dfrac{g}{v^2}\theta$
Confused about the $\pm$ sign?
Multidimensional integral involving delta functions
How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$
How can I get smooth curve at the sigmoid function?
Evaluate the integral $\int_{0}^{+\infty}\left(\frac{x^2}{e^x-1}\right)^2dx$
Is the real number structure unique?
What is the algorithm hiding beneath the complexity in this paper?
A continuous function that attains neither its minimum nor its maximum at any open interval is monotone
How can prove this $\binom{n}{p}\equiv \left\lfloor\frac{n}{p}\right\rfloor \pmod {p^2}$
Find $E(XY)$ assuming no independence with $E(X) = 4$, $E(Y) = 10$, $V(X) = 5$, $V(Y) = 3$, $V(X+Y) = 6$.
Why are the solutions of polynomial equations so unconstrained over the quaternions?
What would be a good way to memorize theorems about algebra?
Is Russell's paradox really about sets as such?
Let $D$ be a UFD. If an element of $D$ is not a square in $D$ then is it true that it is not a square in the fraction field of $D$?