MHB Why is this polynomial separable?

[mathmari]

Hey!

In my notes there is the following:

Let $F$ be a field. The irresducible $f\in F[x]$ is separable, if all the roots are different.
A non-constant polynomial $f\in K[x]$ is separable, if all the irreducible factors are separable.

Example:
$f(x)=(x^2-2)^2(x^2+3)\in \mathbb{Q}[x]$.
The irreducible factors are:

• $x^2-2$ $\rightarrow$ Roots : $\pm 2$ different
• $x^2+3$ $\rightarrow$ Roots : $\pm i\sqrt{3}$ different

So, $f$ is separable. I haven't understood why $f$ is separable. Since we have the factor $(x^2-2)^2$ aren't the roots $\pm 2$ of multiplicity $2$ ? (Wondering)

 

[I like Serena]

Hey!

In my notes there is the following:

Let $F$ be a field. The irresducible $f\in F[x]$ is separable, if all the roots are different.
A non-constant polynomial $f\in K[x]$ is separable, if all the irreducible factors are separable.

Example:
$f(x)=(x^2-2)^2(x^2+3)\in \mathbb{Q}[x]$.
The irreducible factors are:

• $x^2-2$ $\rightarrow$ Roots : $\pm 2$ different
• $x^2+3$ $\rightarrow$ Roots : $\pm i\sqrt{3}$ different

So, $f$ is separable. I haven't understood why $f$ is separable. Since we have the factor $(x^2-2)^2$ aren't the roots $\pm 2$ of multiplicity $2$ ? (Wondering)

Hey mathmari!

Aren't the roots $\pm \sqrt 2$? (Wondering)

And isn't $f(x)$ reducible? That would mean that the first statement doesn't apply.
I think we should apply the second statement first. (Thinking)

 

[mathmari]

Aren't the roots $\pm \sqrt 2$? (Wondering)

Ah yes... (Blush)

And isn't $f(x)$ reducible? That would mean that the first statement doesn't apply.
I think we should apply the second statement first. (Thinking)

Yes, we apply the second statement first. The irreducible factors $x^2-2$ and $x^2+3$, right? (Wondering)

 

[mathmari]

So, an inseparable polynomial is for example $f(x)=(x^2-4)^2(x^3-8)$, where the irreducible factors are $x^2-4$ and $x^3-8$, and the roots are $\pm 2$ and $2$ ? (Wondering)

 

[I like Serena]

Yes, we apply the second statement first. The irreducible factors $x^2-2$ and $x^2+3$, right?

Yes. That's what your example states as well.

So, an inseparable polynomial is for example $f(x)=(x^2-4)^2(x^3-8)$, where the irreducible factors are $x^2-4$ and $x^3-8$, and the roots are $\pm 2$ and $2$?

Can't we reduce $x^3-8$ to $(x-2)(x^2+2x+4)$? (Wondering)

 

[mathmari]

Can't we reduce $x^3-8$ to $(x-2)(x^2+2x+4)$? (Wondering)

Ah, so it is again separable, right? (Wondering)

Could you give me an example of an inseparable polynomial? (Wondering)

 

[I like Serena]

Could you give me an example of an inseparable polynomial? (Wondering)

How about $x^3-2$ in $F_3[x]$? (Wondering)

 

[mathmari]

How about $x^3-2$ in $F_3[x]$? (Wondering)

I saw now in my notes the following proposition:
If $F$ is finite, then each non-constant $f\in F[x]$ is separable.

We have that $F_3$ is finite, right? Therefore, $x^3-2$ is separable, isn't it? (Wondering)

 

[I like Serena]

I saw now in my notes the following proposition:
If $F$ is finite, then each non-constant $f\in F[x]$ is separable.

We have that $F_3$ is finite, right? Therefore, $x^3-2$ is separable, isn't it? (Wondering)

Well... $x^3-2$ is irreducible, but we also have $x^3-2=(x+1)^3$ in $F_3[x]$, so $2$ is a triple root.
If that doesn't mean it's inseparable, I wouldn't know what is.

Do your notes give an example of an inseparable polynomial?
Or do you have a proof for that proposition? (Wondering)

 

[mathmari]

Well... $x^3-2$ is irreducible, but we also have $x^3-2=(x+1)^3$ in $F_3[x]$, so $2$ is a triple root.
If that doesn't mean it's inseparable, I wouldn't know what is.

Do your notes give an example of an inseparable polynomial?
Or do you have a proof for that proposition? (Wondering)

In this book the chapter 3.4 (p.11) is about separability. This proposition is the 3.4.5 and the only example of an inseparable polynomial that I found is the one that is at 3.4.8.

(Thinking)

 

[I like Serena]

In this book the chapter 3.4 (p.11) is about separability. This proposition is the 3.4.5 and the only example of an inseparable polynomial that I found is the one that is at 3.4.8.

(Thinking)

This book gives example 1.2 with $x^3-2$ in $F_3[x]$ as an inseparable polynomial.
But its definition of inseparable seems to be different from your definition.
Btw, wiki seems to give a different definition as well. (Thinking)

I can see that your book considers $x^3-2$ reducible, since it is reducible over $F_3$. It seems that my book reference considers $x^3-2$ irreducible, but to be honest I'm not sure why. Wiki's definition of irreducible seems to imply that it's reducible.