-4
votes
3answers
104 views

What is the cardinality of free product $\mathbb{Z} * \mathbb{Z}$? [closed]

I want to know cardinality of $\mathbb{Z} * \mathbb{Z}$. Is it countable? or uncountable?
0
votes
0answers
66 views

Can I assign a distinct homomorphism to every function?

I consider an algebraic structure and particular structure preserving morphisms (think groups and group homomorphism). I wonder if I can assign a distinct such morphisms to every function between any ...
-1
votes
2answers
179 views

Cardinality of algebraic extensions of an infinite field.

An exercise in Lang's algebra book is: let $k$ an infinite field, and $E$ an algebraic extension of $k$. Then $E$ has the same cardinality as $k$. How can one can prove this?
2
votes
1answer
90 views

Cardinality of the quotient field of power series ring

Let $k$ be a field which is countable and let $x$ be an indeterminate over $k$. I have hard time to prove $$\operatorname{card} k((x)) = \operatorname{card}\mathbb R.$$ Thank you.
1
vote
1answer
112 views

Dimension of $\operatorname{Hom}(V, W)$

What is the dimension of $\operatorname{Hom}(V, W)$ if at least one of the two vector spaces $V, W$ is infinite dimensional? In the sense of cardinal numbers. Thanks
2
votes
1answer
170 views

Can we embed the ordinal and cardinal number systems into larger, more convenient systems of arithmetic?

We can embed $\mathbb{N}$ in a larger number system, such as $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$, for convenience. Now since $\mathbb{N}$ is extended by $\mathrm{Ord}$ and $\mathrm{Card}$, the ...
2
votes
2answers
156 views

What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$?

What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$? Is it that of the continuum? Proof?
6
votes
0answers
129 views

Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without ...