2
votes
3answers
223 views

Upper bound on cardinality of a field

Is there an upper bound on the cardinality of a field? The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...
2
votes
1answer
40 views

Uncountable, algebraically independent subset of $\mathbb{C}$?

Does such a subset exist? I am interested in algebraic independence over $\mathbb{Q}$. Could this be proven in an abstract way or would it be more appropriate to construct an explicit example?
0
votes
1answer
24 views

countable set that contains 1 and pi and has polynomial with coefficients in set s.t. all real roots are in set

Deduce that there is a countable set X that contains the real numbers 1 and pi and has the further property that if P is any non-zero polynomial with coefficients in X, then all real roots of P belong ...
1
vote
5answers
130 views

How can one find a set of given cardinality and disjoint from a given set?

In Algebra by Serge Lang, the author asserts, to prove the existence of a field extension where an irreducible polynom has a root, that if you take one set $A$ and a cardinal $\mathcal{C}$, that you ...
1
vote
1answer
58 views

Set of Functions is a Vector Space problem

Let $F$ be a field. Consider the $F$-vector space $F(X, F)$ of all functions from $X = \{ 1, 2 \} \to F$. Define $e_1, e_2 \in F(X, F)$ by $e_1 = \{ (1, 1),(2, 0) \}$ and $e_2 = \{ (1, 0),(2, 1) \}$. ...
4
votes
2answers
163 views

Is there an uncountable proper subfield of $\mathbf{R}$?

Whether there is a uncountable proper subfield of real line $\mathbf{R}$?? Thanks a lot!
5
votes
1answer
119 views

bijection from $\mathbb{Q} - \{a\}$ to $\mathbb{Q}$ using elementary functions only?

I was wondering, can you define a bijection from $\mathbb{Q} - \{a\}$ to $\mathbb{Q}$ using elementary functions only ($a \in \mathbb{Q}$)? Of course there are many set theoretic bijections like ...
1
vote
1answer
54 views

Ordered Field/Ordered Set Question

I have 2 statements that I need to say whether they are True or False. I do not need a proof. I would like some confirmation whether my answers are correct. Every ordered field has the least upper ...
1
vote
1answer
59 views

Build Onto mapping of a Field to a field with an Integral Domain

The question is as follows, Let D be an integral domain. Let Q be its field of fractions and $\phi$ : D --> Q be the canonical map of D into Q. Prove that, if D is a field, then $\phi$ is ...
2
votes
2answers
146 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?
1
vote
2answers
244 views

When are quotient maps induced by equivalence relations surjective and injective?

Let $\sim$ be an equivalence relation on a set $X$. Also, there is a natural function $p:X\to \tilde X$ where $\tilde X$ is a set of all equivalence classes. (Equivalence classes are defined as, ...
2
votes
2answers
113 views

Why does $\# F(X)=\max(\# F,\# X)$ when $\# X$ is a transcendence basis?

I recently asked a question about the transcendence degree of $\mathbb{C}$ over $\mathbb{Q}$. In a nice answer which I hope to understand better, two facts were given: 1: If $F$ is any infinite field ...
5
votes
2answers
199 views

How many fields inside $\mathbb R$?

i.e. what is cardinality of $\{A \mid \ A\subset \mathbb R, A \text{ is a field} \}$?
2
votes
2answers
972 views

inverse element in a field of sets

A field is an abelian group under addition and a group under multiplication. But for the power set of a set S, under union and intersection operations and with S and the empty set being their ...
2
votes
1answer
764 views

Field of sets and Sigma algebra of sets

(1). According to Wikipedia, a field of subsets of X is defined to be a non-empty subset of the power set of X closed under the intersection and union of pairs of sets and under complements of ...