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### Field Theory: Splitting Field

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

Hello, Dr. Math.

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,

Lucia
```
Associated Topics:
College Modern Algebra

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