Field Theory: Splitting Field

Date: 06/12/2002 at 23:31:39
From: Lucia
Subject: Field theory-Abstract algebra

Hello, Dr. Math.

Thank you for the help you have given me in the past. Can you please 
help with the following:

  Find the splitting field of (x^3 - 5)

Thank you very much.
Lucia

Date: 06/13/2002 at 11:58:06
From: Doctor Paul
Subject: Re: Field theory-Abstract algebra

What are the three roots?  Well, you know that the real root is 
cbrt(5).  But there are two imaginary roots.  

Notice that if zeta_3 is a primitive third root of unity (i.e., 

  (zeta_3)^3 = 1 

and no smaller power of zeta_3 equals one) then

  [cbrt(5) * zeta_3]^3 - 5 = 5*1 - 5 

                           = 0

Thus cbrt(5) * zeta_3 is also a root of x^3 - 5.

Also notice that 

  [cbrt(5) * (zeta_3)^2]^3 - 5 = 5*(zeta_3)^6 - 5 

                               = 5*[(zeta_3)^3] ^2 - 5 

                               = 5*1^2 - 5 = 5*1 - 5 

                               = 0

Thus cbrt(5) * (zeta_3)^2 is also a root of x^3 - 5.

There can be only three roots, so we have found them all.  The
splitting field is found by adjoining all of the roots to Q.  

Thus the splitting field is:

  Q(cbrt(5), cbrt(5) * zeta_3, cbrt(5) * (zeta_3)^2)

This isn't a very pretty way of representing the splitting field, but 
it is correct.

Let 

  F = Q(cbrt(5), cbrt(5) * zeta_3, cbrt(5) * (zeta_3)^2)

It is in fact the case that 

  F = Q(cbrt(5), zeta_3).

And seeing this is really not hard at all.

F is certainly contained in Q(cbrt(5), zeta_3) since fields are closed 
under multiplication.

To see that Q(cbrt(5), zeta_3) is contained in F, we need to write 
cbrt(5) and zeta_3 as a combination of field operations involving only 
elements from F.

This is quite easy:  

  cbrt(5) = cbrt(5) 

and 

  zeta_3 = cbrt(5) * zeta_3 / cbrt(5).

This shows inclusion in both directions and establishes the fact that 
F = Q(cbrt(5), zeta_3).

Thus the splitting field is Q(cbrt(5), zeta_3).

I hope this helps.  Please write back if you'd like to talk about 
this some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/13/2002 at 16:51:21
From: Lucia
Subject: Thank you (Field theory-Abstract algebra)

Dear Dr.Paul,

Thank you for your complete and easy-to-understend answer. It was very
helpful.

Lucia