quadratic expansion generating the field of real numbers

Asked Jan 23 '25 at 07:06

Active Jan 23 '25 at 07:06

Viewed 37 times

Let R be the field of real numbers. Is there any extension of fields K ⊂ R such that [R : K] = 2?

It is known that the field of real numbers is an extension of the field of rational numbers; the degree of this expansion is equal to the power of the continuum, so this expansion is infinite.

This gives rise to the hypothesis that the desired extension does not exist. Tell me how to prove this or, if I am wrong, how to construct such an extension?

asked Jan 23 '25 at 07:06

aviaf

1

See this old answer of mine for a clue on how to prove that no such $K$ can exist. Originally that question asked for a field $K$ such that $\Bbb{R}=K(\sqrt2)$, but it was later edited to a more difficult version. Not sure I want to vote close this as a duplicate (the edit being a good reason). – Jyrki Lahtonen Jan 23 '25 at 07:10
Okay, thanks for clarifying, @Jyrki Lahtonen – aviaf Jan 23 '25 at 07:24

quadratic expansion generating the field of real numbers

0 Answers0