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Let

Then,

1. Is

Or, simply, is there a polynomial

with rational coefficients such that2. What if

is replaced with ?'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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Yes, and there are two inclusions you need to prove. The inclusion is clear. For the reverse inclusion, note that has inverse So you know now that both and are elements of the ring , and so any -linear combination of these also belongs to the same ring. Deduce that must therefore belong to the ring (as some linear combination of those two elements) and do the same for . This proves the reverse inclusion, and thus, these rings are in fact the same. (By the way, if you've done any Galois theory, you could also use the fact that since both radicals are trivially algebraic over , and then use the tower law.)For the second question, try assuming that and derive a contradiction.

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