Tagged Questions

4
votes
2answers
129 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
59 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
185 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. ...
5
votes
3answers
401 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
101 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
145 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
195 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 ...