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Part BCSIR NET December 2023dense-subfields

Dense subfields

Consider the field C\mathbb{C} with the Euclidean topology. Let K be a proper subfield of C\mathbb{C} that is not contained in R\mathbb{R}. Which one of the following statements is necessarily true?

  1. A.K is dense in C\mathbb{C}.✓ Correct
  2. B.K is an algebraic extension of Q\mathbb{Q}.
  3. C.C\mathbb{C} is an algebraic extension of K.
  4. D.The smallest closed subset of C\mathbb{C} containing K is NOT a field.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests finite-dimensional intuition.

See pricing

50 are analysed free — try those first.

The trap it tests

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Drill statements like this

Related counterexample: An algebraic extension is a finite extension

More on this topic

The chapter behind this: Field extensions, degrees and splitting fields — free to read

From Rings and Fields › Field extensions, splitting fields, finite fields

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