Question

(a)If F has characteristic 0, $f\left(x\right)\in F\left[x\right]$ and $f\left(x\right)=0_F$ prove that $f\left(x\right)=c$ for some $c\in F$.

(b)Give an example in $Z_2\left[x\right]$ to show that part a may be false if F does not have characteristic 0.

Short answer

(a) It is proved that $f\left(x\right)=c$

(b)Result of part a is false if $\text{char}F\ne 0$

Step-by-step solution

Definition of polynomial

A polynomial $p\left(x\right)$ over a given field k is separable if its root are distinct in an algebraic closure of k .

(a) Step2: Showing that fx=c

If F has characteristic 0, $f\left(x\right)\in F\left[x\right]$and $f^{\text{'}}\left(x\right)=0$. Let $f\left(x\right)=a_0+a_{1}x+......+a_n{x}^n\in F\left[x\right]$then $f^{\text{'}}\left(x\right)=a_1+......+n{a}_n{x}^{n-1}$as $f^{\text{'}}\left(x\right)=0$. So this gives:

$a_1+......+n{a}_n{x}^{n-1}=0$

This implies $i{a}_i=0,i=1,2,3,.......n$ also as $charF=0$ therefore $a_i=0,i=1,2,3....n$ thus this implies that f is constant function.

Therefore $f\left(x\right)=c$ for some $c\in F$.

(b) Step 3: Showing that result of part a is false if charF≠0

Consider the polynomial $f\left(x\right)=x^2+1$over $Z_2\left[x\right]$ now differentiate the polynomial.

$$\begin{gathered} f\left(x\right)=x^2+1 \\ {f}^{\text{'}}\left(x\right)=2x \end{gathered}$$

Therefore the part a is false if $\text{char}F\ne 0$.

The answer is:

(a) It is proved that$f\left(x\right)=c$

(b)Result of part a is false if$\text{char}F\ne 0$.