Approaching Question C1: Finding Roots of Elements in a Field

[Kiwi1]

View attachment 5969

Any suggestions on how I should approach question C1?

Every time I think I have a solution I find that I have made the implicit assumption either that F is abelian or that the roots of w are in the center of F. I don't think either assumption is valid.

If I let K be the root field of the poly then clearly it must contain d, but I have been unable to show that it must contain w.

I can see that:
\([F(\omega):F] \leq p-1\)

\([F(d):F] \leq p\)

\([F(d,\omega):F]=[F(d,\omega):F(d)][F(d):F]\)

\([F(d,\omega):F]=[F(d,\omega):F(\omega)][F(\omega):F]\)

But don't seem to be able to form these ideas into a solution.

 

[Kiwi1]

OK so solved Q1. It was straightforward to show (by contradiction) that w is in the center of K. From there it is easy enough to show that \(F(d,\omega)\) is the root field.

But now I can't solve Q2. In fact I can falsify the assertion of Q2 as follows:

Let F be the field of rational numbers.

\(x^5-2^5=(x-2)(x^4+2x^3+4x^2+8x+16)\)

So with p=5 and a=32 I get:
\(x^p-a=f(x)p(x)\) where f(x) has degree 1 and p(x) has degree 4

 

[caffeinemachine]

OK so solved Q1. It was straightforward to show (by contradiction) that w is in the center of K. From there it is easy enough to show that \(F(d,\omega)\) is the root field.

But now I can't solve Q2. In fact I can falsify the assertion of Q2 as follows:

Let F be the field of rational numbers.

\(x^5-2^5=(x-2)(x^4+2x^3+4x^2+8x+16)\)

So with p=5 and a=32 I get:
\(x^p-a=f(x)p(x)\) where f(x) has degree 1 and p(x) has degree 4

I think the question is asking to show this:

We can factor $x^p-a$ as $f(x)p(x)$ where both factors have degree at least $2$. (Note that the question says "at most" in place of "at least", but this is clearly a typo since then deg(x^p-a)=p\leq 4).

 

[Kiwi1]

Once we start to speculate on what a typo is things get hard. Perhaps they mean ONE factor is \(\leq 2\). The example I have shown has one factor of degree \(\leq 2\) and one \(\geq 2\).

Looking at the subsequent questions it would make more sense to me that they would want me to prove ONE factor has degree greater than 2. In the context of the rational numbers that would ensure that the extended field includes a complex number.

Having completed questions C3-C7, I can't see why question C2 is required to complete the proof that: \(x^p-a\) either has a root in F or is irreducible over F.

 

[caffeinemachine]

Once we start to speculate on what a typo is things get hard. Perhaps they mean ONE factor is \(\leq 2\). The example I have shown has one factor of degree \(\leq 2\) and one \(\geq 2\).

Looking at the subsequent questions it would make more sense to me that they would want me to prove ONE factor has degree greater than 2. In the context of the rational numbers that would ensure that the extended field includes a complex number.

Having completed questions C3-C7, I can't see why question C2 is required to complete the proof that: \(x^p-a\) either has a root in F or is irreducible over F.

I think you are right. Anyway, I don't think this question is worth the time. :)