Articles of abstract algebra

Finite Extensions and Bases

The question reads: Find a basis of $\mathbb{Q}(\sqrt{2}+\sqrt[3]{4})$ over $\mathbb{Q}$. Describe the elements of $\mathbb{Q}(\sqrt{2}+\sqrt[3]{4})$. My original thought on the approach was to find the minimum polynomial (which would have degree 6), and then just take create a basis where every element is $(\sqrt{2}+\sqrt[3]{4})^n$ for $n=0,\ldots,5$. The hint in the back recommends adjoining $\sqrt[3]{4}$ first, […]

$X$ a set and $G$ a group. Let $G^X$ be the set of mappings from $X$ to $G$, show that $G^X$ can have same structure.

So I’ve got this exercise: Let $X$ be a set and $(G, \star)$ a group. We denote $G^X$ the set of a mappings from $X$ to $G$. Show that $G^X$ has a group structure induced by $G$ So this is my attempt. I first have to show that it has the identity element, but I […]

Basic question regarding a finitely generated graded $A$-algebra

Let $S = \oplus_{n \geq 0} S_n$ be a graded ring. Let $S$ be a finitely generated $A$-algebra, where $A = S_0$, a commutative ring with unity. Then there exists $t_1, .., t_M$ homogeneous elements of positive degree that generate $S$ over $A$. It then follows that $S$ is isomorphic to $k[x_1, …, x_M]/I$ as […]

if the kernel of an endomorphism is a direct summand, doesn't the image have to be too?

Let $M$ be a module and $\phi: M\longrightarrow M$ be an endomorphism. Let $\ker \phi$ be a direct summand of $M.$ Does it imply that $\mbox{im}\,\phi$ is a direct summand of $M$ too? I have not seen an explicit statement in any text have read but it seems quite clear from what I have seen […]

Necessary and Sufficient Condition for $\phi(i) = g^i$ as a homomorphism – Fraleigh p. 135 13.55

Let $g \in \text{ group } G $ and $n \in N$. Let $\phi : \mathbb{Z_n} \rightarrow G$ be defined by $\phi(i) = g^i$ for $0 \le i \le n$. Give a necessary and sufficient condition (in terms of g and n) for $\phi$ to be a homomorphism. Prove your assertion. My $g =$ solution’s […]

zero divisors of ${\bf Z}_n$

Consider the following proposition: A nonzero element $m\in{\bf Z}_n$ is a zero divisor if and only if $m$ and $n$ are not relatively prime. I don’t know if this is a classical textbook result. (I didn’t find it in Gallian’s book). For the “only if” part, one may like to use the Euclid’s lemma. But […]

$S_n$ acting transitively on $\{1, 2, \dots, n\}$

I am reading Dummit and Foote, and in Section 4.1: Group Actions and Permutation Representations they give the following example of a group action: The symmetric group $G = S_n$ acts transitively in its usual action as permutations on $A = \{1, 2, \dots, n\}$. Note that the stabilizer in $G$ of any point $i$ […]

Existence of the least common multiple in a Unique Factorization Domain

Let $x,y \in R$, where $R$ is a unique factorization domain. Let $P$ be the set of all representatives in each class of associate irreducible elements of $R$. Then, suppose $x,y\in R$ are nonzero, nonunit elements (First Question: How would this proof change if either $x$ or $y$ is zero or is a unit of […]

Norm of powers of a maximal ideal

Let $A$ be a integral domain and $M$ a maximal ideal in $A$ such that the quotient $A/M$ is a finite ring (and thus a finite field). Is it true, in general, that $$|A/M^k|=|A/M|^k \quad (k\in\textbf{N})\ ?$$ Edit. (Counter-example in the answers, thanks to Jendrik Stelzner and Bib-lost). Nevertheless, I have the feeling that this […]

Integral Domains

If $R$ is an integral domain and a subring $S$ has identity $1_S$, how would you show that $1_S=1_R$ (here $1_R$ is the identity of the ring $R$)? I am unsure about what an integral domain really is and how the subring comes into play here.