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Field Extensions of the Rationals

Hello all,

Is every field extension of the rationals a subfield of the complex numbers?

Failing that, is every finite field extension of the rationals (algebraic number field) a subfield of the complex numbers?

(I do not know what the degree of the complex numbers over the rationals is!)

Proofs or sketch-proofs would be much appreciated!

Thank you :smile:

Reply 1

Smaug123

Original post by Magu1re

Is every field extension of the rationals a subfield of the complex numbers?

Extend to the p-adic numbers. Because the metric is now different (p-adic metric, not Euclidean metric), you get things which aren't really complex at all - approaching infinity (wrt the Euclidean metric) in different ways gives you different p-adic numbers. I'm not sure if that's quite what you were looking for, though.