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#1 2017-08-25 16:37:22

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
Website

Are these two rings equal?

Let

Then,

1. Is

Or, simply, is there a polynomial

with rational coefficients such that

2. 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'
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#2 2017-08-27 07:03:27

zetafunc
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Registered: 2014-05-21
Posts: 2,107
Website

Re: Are these two rings equal?

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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