1
vote
1answer
66 views

Show that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic to the field $\mathbb{Q}$

I have problem in showing that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic as a ring to the field $\mathbb{Q}$. Any idea? Thanks
2
votes
1answer
63 views

Structure of a module and finite fields

I am trying the following problem which appeared on an exam for a grad course: Determine the structure of finite field $F$ as a module over $F_p[x]$ when $xv=Lv$. Here $L(z)=z^p$ and ...
-2
votes
4answers
85 views

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$. How to prove? I really have no idea... Thank you a lot.
0
votes
1answer
37 views

field extension $F(x)=F(x^2)$ [duplicate]

Let $x$ be algebraic over $F$ such that the field extension $F(x):F$ satisfies $[F(x):F]$ odd. Then prove $[F(x):F]=[F(x^2):F]$ hence $F(x)=F(x^2)$. How to prove? I only obtained the proof for the ...
6
votes
2answers
346 views

Counterexamples in algebra

I got the feeling that whenever a subject gets so sophisticated that Zorn's lemma is needed, a book of counterexamples in that subject would probably benefit researchers/ students a lot. Zorn's ...
2
votes
1answer
108 views

Injection from an integral domain to its field of fractions.

I have a quick question about modules. Suppose that $R$ is an integral domain with field of fractions $K$. Then any free $R$-module is isomorphic to copies of direct sums of $R$, say $R^i$ . ...
8
votes
2answers
272 views

The Noether-Deuring Theorem

I have to solve the following exercise taken from the book "Introduction to Representation Theory" by P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina and S. ...
7
votes
3answers
997 views

If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. [duplicate]

Possible Duplicate: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I need to prove this result, but the only starting point I think of is to ...
2
votes
1answer
125 views

Field homomorphism $\varphi:\mathbb{Q}(\sqrt[3]{2})\to\mathbb{K}$ where $\mathbb{K}$ is the splitting field of $x^3-2$

Denote $\mathbb{K}$ as the splitting field of $x^3-2$ . I wish to find $\Gamma:=\left\{ \varphi:\mathbb{Q}(\sqrt[3]{2})\to\mathbb{K}|\forall q\in\mathbb{Q}:\varphi(q)=q\right\} $. Sind $\varphi$ is ...
2
votes
1answer
149 views

A question with an odd hypothesis.

Let $S$ be a discrete valuation ring and $R\subset S$ be a proper subring (also a DVR). Assuming that $M$ and $N$ are the respective maximal ideals of $R$ and $S$ and that $N\cap R = M$, then the ...
3
votes
2answers
218 views

Extending morphisms with Zorn's lemma

I have stumbled upon a remarkable similarity between the proof of Baer's criterion and an extension theorem in field theory. Here are the statements: Baer's criterion: Let $R$ be a ring. A left ...