Math Is Fun Forum

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)

laipou

Member

Registered: 2009-09-19

Posts: 24

Re: radical extention

Sorry,the title should be "radical extension".

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..."

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)

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..."

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.