5
votes
1answer
143 views

Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...
1
vote
2answers
75 views

Is a set that is an abelian group under addition and a group under multipliation a field?

I suspect the answer to my question is yes, but I'm just checking my understanding. If we have a set which is an Abelian group under addition and a group under multiplication is it then defined as a ...
3
votes
1answer
127 views

When is a divisible group a power of the multiplicative group of an algebraically closed field?

It is known that for any algebraically closed field $\mathbb{F}$ its multiplicative group $\mathbb{F}^*$ is a divisible group, and consequently any power $\mathbb{F}^*\times\cdots\times \mathbb{F}^*.$ ...
4
votes
1answer
45 views

Group automorphisms of the non-zero elements of a field

I would like to know what is $Hom_{Groups}(K^*,K^*)$, at least in the case $K$ is a complete non-archimedean (valued) field. Is this $\mathbb{Z}$?
1
vote
1answer
61 views

Fields arising as endomorphism rings

Do you know a field $K$ other than $F_p$ which is the endomorphism ring of an abelian group $G$? I doubt that there is one because as $G$ gets bigger, $End(G)$ seems to be more and more ...
0
votes
2answers
106 views

Relation between torsion subgroup of multiplicative group of field and solvability of polynomials

In a broad sense, what relationships are there between the torsion subgroup $G$ of the multiplicative group of non-zero elements of a field $K$ and whether or not certain polynomials in $K[x]$ have ...
2
votes
1answer
95 views

number of differents vector space structures over the same field $\mathbb{F}$ on an abelian group

My question here raised another one. How many differents vector space structures over a field $\mathbb{F}$ we may have on an abelian group? I know that there are abelian groups that we can not endow ...