Math Is Fun Forum

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

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

?

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zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

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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.