Instead of extending a field, by adjoining a new element, consider what happens if we remove an element or elements.

This started as a question on math.SE Field reductions where Pete L. Clark explained that there isn't a unique subfield that could be called a "field reduction", rather there is a set of maximal subfields not containing the element.

Let $\mathbb{R}(\setminus a)$ be the set of maximal subfields of $\mathbb{R}$ that don't contain $a$.

Question: What is the cardinality of $\mathbb{R}(\setminus a)$ ?

Pete L. Clark suggested that this sounds "ultrafiltery" so I thought it would be an appropriate question for Mathoverflow.

Let $\mathbb{A}'$ be the set of non-rational algebraic numbers.

In Field reductions. part two Arturo Magidin showed that the cardinality of $\mathbb{R}(\setminus\mathbb{A}')$ is $2^\mathfrak{c}$.