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Integral Domains, Fields and Subfields

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Problem:
Note: Q is rational numbers, R is real numbers ,
sqrt() means square root

Show that Q(sqrt(2)) is the smallest subfield of R that contains sqrt(2).

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Solution Summary

A proof involving rational and real numbers is provided.

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Proof:
First, Q(sqrt(2)) is a subfield of R. An arbitrary element in Q(sqrt(2)) has the form x=a+bsqrt(2) with a,b in Q. We note ...

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