Coefficient of a polynomial in K[x] where K is a field of characteristic p

[rukawakaede]

Let K be a field of characteristic p.

Suppose f(x)=(x^k+c_k-1x^k-1+...+c₀)(x^p-k+...) in K[x] with 1≤k≤p-1.

My question is:

1. since f(x) in K[x], can I conclude g(x)=x^k+c_k-1x^k+...+c₀ in K[x] as well?

2. We see that in general if g(x)=x^k+c_k-1x^k-1+...+c₀ then c_k-1=-(α₁+α₂+...+α_k) where α₁,α₁,...,α_k are the root of g(x).

Now if c_k-1∈K, can I conclude that α₁,α₁,...,α_k∈K as well?

very appreciated if someone could solve my confusion.
Thank you!

 

[micromass]

Hi rukawakaede!

I'm not sure where these questions come from, so I don't know if I interpreted them correctly:

Let K be a field of characteristic p.

Suppose f(x)=(x^k+c_k-1x^k+...+c₀)(x^p-k+...) in K[x] with 1≤k≤p-1.

My question is:

1. since f(x) in K[x], can I conclude g(x)=x^k+c_k-1x^k+...+c₀ in K[x] as well?

Take $K=\mathbb{F}_2$ and take f(X)=X²+X+1. Then f splits in a field extension of K, that is, we have

$$f(X)=(X-\alpha_1)(X-\alpha_2)$$

but we won't have that $(X-\alpha_i)$ in K[X] (otherwise $\alpha_i\in \mathbb{F}_2$ which cannot be).

2. We see that in general if g(x)=x^k+c_k-1x^k+...+c₀ then c_k-1=-(α₁+α₂+...+α_k) where α₁,α₁,...,α_k are the root of g(x).

Now if c_k-1∈K, can I conclude that α₁,α₁,...,α_k∈K as well?

No, see my previous example. Another example which is not in char 2, is the polynomial $X^2+1$ in $\mathbb{R}$. The sum of the roots is i-i=0, but neither i nor -i are in $\mathbb{R}$...

 

[rukawakaede]

Hi rukawakaede!

I'm not sure where these questions come from, so I don't know if I interpreted them correctly:
Take $K=\mathbb{F}_2$ and take f(X)=X²+X+1. Then f splits in a field extension of K, that is, we have

$$f(X)=(X-\alpha_1)(X-\alpha_2)$$

but we won't have that $(X-\alpha_i)$ in K[X] (otherwise $\alpha_i\in \mathbb{F}_2$ which cannot be).
No, see my previous example. Another example which is not in char 2, is the polynomial $X^2+1$ in $\mathbb{R}$. The sum of the roots is i-i=0, but neither i nor -i are in $\mathbb{R}$...

Thanks micromass!

I understand what you said.

In fact the polynomial is given as f(x)=x^p-x-a where a∈K.

I want to show that c_k-1∈K implies f(x) has a root in K but stuck here.

Since p=2 case is trivial. does that means we can show this by induction?Could you please give some suggestions?

 

[micromass]

Ok, rukawakaede, the polynomials you mention are called "Artin-Schreier polynomials" and are very well studied.

In general, if we have the polynomial $X^p-X+\alpha$ and if $\beta$ is one root of the polynomia (in some field extension of K)l, then we know all the roots:

$$\beta,\beta+1,...,\beta+p-1$$

this is essentially Fermat's little theorem. So let

$$L=K[x]/(X^p-X+\alpha)$$

then L is the field extension which contains all the roots of the polynomial. Can you solve your question now? Tell me if you need more hints...

 

[rukawakaede]

Ok, rukawakaede, the polynomials you mention are called "Artin-Schreier polynomials" and are very well studied.

In general, if we have the polynomial $X^p-X+\alpha$ and if $\beta$ is one root of the polynomia (in some field extension of K)l, then we know all the roots:

$$\beta,\beta+1,...,\beta+p-1$$

this is essentially Fermat's little theorem. So let

$$L=K[x]/(X^p-X+\alpha)$$

then L is the field extension which contains all the roots of the polynomial. Can you solve your question now? Tell me if you need more hints...

I know if K adjoint a root of f(x), called it α, then in fact K(α) consists all of its roots, i.e. K(α,α+1,...,α+(p-1))=K(α) and so K(α) is the splitting field of f(x).

Now I have question:
if c_k-1, as defined above, is in K, then we said the polynomial f(x) splits in K[x]?I am not fully understand, in particular, how can I show the implication c_k-1∈K implies f(x) has a root in K?

 

[rukawakaede]

micromass thanks!
I am now fully understood.
Thank you very much, especially for telling me the name of the polynomial. Learned extra stuff! :)