So, what are the subfields of C that are isomorphic to R?

You can make some of them this way: Choose a maximal algebraically independent subset of the reals, call it S. "Algebraically independent" means that no non-trivial polynomial over the algebraic numbers is 0.

Then, if we choose another algebraically independent subset T of the reals,
of the same cardinality as S, then any bijection

The algebraic closure of Q(S), cl(Q(S)), must contain R, so it is C.

What's the size of S? Since C has cardinality 2^{ℵ} and the
cardinality of an infinite field is the same as the cardinality of its
algebraic closure, Q(S) must have cardinality 2^{ℵ}.
So S has cardinality 2^{ℵ}.

So, the cardinality of isomorphisms of Q(S) to Q(T) is
2^{2ℵ}, since
there are 2^{2ℵ} bijections between S and T.

Any isomorphism of Q(S) to Q(T) can be extended to an isomorphism of

Suppose f, g are distinct isomorphisms ^{ -1} g

So, there are 2^{2ℵ} subfields of C isomorphic
to R. And also 2^{2ℵ}subfields of C of index 2
in C, isomorphic to R!
The cardinality
of the set of such subfields can't be any greater than 2^{2ℵ}, since
that is also the cardinality of subsets of C.

But might there be subfields of C isomorphic to R, that aren't obtainable this way? This method produces a field isomorphic to R with an uncountable intersection with R.

But you can also find fields isomorphic to R that intersect R only in the
algebraic numbers. First, split S, the maximal algebraically independent
subset of the reals, into two sets X and Y of
the same cardinality. Let f be a bijection

__Proof:__ α is also algebraic over

The sets

α is algebraic over

There's a lemma in Lang's Algebra which says: Let K be algebraically closed in extension L. Let α be some element of an extension of L, but algebraic over K. Then [K(x):K] = [L(x):L].

This implies that if you have a field K, and S_{1} and
S_{2} are transcendental over K and algebraically independent of
each other, then _{1}) ∩ cl(K(S_{2}) =
cl(K)._{1}), then this polynomial is also the minimal polynomial for
α over _{1}, S_{2})._{2}).
This means that the minimal polynomial for α over
_{1}, S_{2}).

Suppose that f(R) is of index 2 in C. Since R is of index 2 in C, is it possible
to say anything about the index of

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