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#1 2010-04-23 16:18:56

laipou
Member
Registered: 2009-09-19
Posts: 24

radical extention

(1)Is there exists a polynomial

such that its splitting field
is contained in a radical extension
over
,but
is not radical over
?
(2)suppose
.
    What does "Solve for a by radicals" mean?

Thanks for any help.

Last edited by laipou (2010-04-23 16:19:22)

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#2 2010-04-24 23:51:48

laipou
Member
Registered: 2009-09-19
Posts: 24

Re: radical extention

Sorry,the title should be "radical extension".

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#3 2010-04-25 02:54:54

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: radical extention

1. If F is a radical extension, then

What can you say about the intermediate fields in between F and Q?

2. "Solve by radicals" means come up with an expression for e^(2pi i)/7 using only rational numbers, addition, subtraction, multiplication, division, and (rational) exponents.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2010-04-25 21:24:25

laipou
Member
Registered: 2009-09-19
Posts: 24

Re: radical extention

All I know is that

is solvable for all intermediate field
.
Is this helpful?

Last edited by laipou (2010-04-25 21:26:47)

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#5 2010-04-26 01:53:30

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: radical extention

Solvable won't do it for you.  Think "Fundamental theorem of Galois theory".


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#6 2010-04-26 03:50:15

laipou
Member
Registered: 2009-09-19
Posts: 24

Re: radical extention

I don't understand how "Fundamental theorem of Galois theory" works here.
Could you tell me more?
Thx.

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