Re: Simple extension field of Q?

On Tue, 27 Nov 2007 21:12:07 +0100, "Manuel Dickenson"
<ask.for.it@xxxxxxxxx> wrote:

Hello, All!

I'm on my way to learn all that gibberish theory on polynomials,
factorization and fields.

Well that's your first mistake.

If you insult the theory, the problems will naturally take their
revenge.

Before getting into the Galois theory, I had one problem
with extension fields.

I have the rational polynomial X^3-7, whose roots are:

a_0 = cubicroot(7)
a_1 = -1/2 + sqrt(3)*i/2
a_2 = -1/2 - sqrt(3)*i/2

Those are not the 3 roots of x^3 - 7.

But worse, did you even state a problem? No. What is the actual
problem? Make sure to show some care by wording it correctly.

Perhaps the problem was to find a single generator for the splitting
field of x^3 - 7 over Q? With that assumption ...

It is true that the splitting field of x^3 - 7 over Q is
Q(cbrt(7),sqrt(-3)).

In general, if a has degree 3 over Q and b has degree 2 over Q, then
a+ b has degree 6 over Q.

Hence the splitting field is Q(c) where c = a + b.

But who knows if my analysis relates to the actual problem. Didn't I
already ask you to state the actual problem?

Well?

quasi
.

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