Does \(\displaystyle\sqrt{{{1}+\sqrt{{{2}}}}}\) belong to \(\displaystyle{\mathbb{{{Q}}}}{\left(\sqrt{{{2}}}\right)}\)?

r1fa8dy5

r1fa8dy5

Answered question

2022-04-05

Does 1+2 belong to Q(2)?

Answer & Explanation

Jermaine Lam

Jermaine Lam

Beginner2022-04-06Added 11 answers

The truth is that 1+2 cannot be expressed as a+b2 where a,b are integers or rationals. If they were reals then the problem becomes trivial.
(Another way to see the first fact above is that the minimal polynomial of 1+2 is degree-4, whereas if it were expressible as a+b2 with a, bQ the minimal polynomial would only be quadratic.)
Aaliyah Phillips

Aaliyah Phillips

Beginner2022-04-07Added 9 answers

With a couple of corrections, your answer seems consistent:
z=d1±i2a=±d1±i2
±i+4b2=14b2=1ib=±d1i2

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