Question

Let K be a finite field of characteristic p, F is a subfield of K, and m is a positive integer. If $\mathbf{L}\mathbf{=}\{\mathbf{a}\in \mathbf{K}|{\mathbf{a}}^{{\mathbf{p}}^m}\in \mathbf{F}\}$, prove that

(a) L is a subfield of K that contains F.

(b)$\mathbf{L}\mathbf{=}\mathbf{F}$[Hint : Use Exercise 10 to show that the map $\mathbf{g}\mathbf{:}\mathbf{K}\to \mathbf{K}$ given by $\mathbf{g}\left(\mathbf{a}\right)\mathbf{=}{\mathbf{a}}^{{\mathbf{n}}^m}$is an isomorphism such that $\mathbf{g}\left(\mathbf{F}\right)\mathbf{=}\mathbf{F}$. What is ${\mathbf{g}}^{-1}\left(\mathbf{F}\right)$]

Short answer

(a) It is proved that L is a subfield of K that contains F.

(b) It is proved that $L=F$

Step-by-step solution

Step 1:Define Homomorphism:

In algebra the Homomorphism is the mapping that defines the structure between the two algebraic structures with the same properties. The two structure can be group, vector spaces or the rings.

Step 2:Prove that L is a subfield of K that contains F.(a)

Assume that the finite field of characteristic p be K and F be a field of K.

Let m be a positive integer.

Assume that $L=\{a\in K|a^{p^m}\in F\}$

Now, to prove that the subfield of K is L that contains F.

Proof:

Let a belongs to L and b belongs to L that is $a\in L$and $b\in L$.

Then $a,b\in K$such that $a^{p^m},b^{p^m}\in F$

Now, to prove that $(a-b)\in L$

As, $a^{p^m}-b^{p^m}={(a-b)}^{p^m}$and also since F is a field,

This suggests that $a^{p^m}-b^{p^m}\in F$

Now , powers are same in the above equation, so taking it common

Therefore, $a^{p^m}-b^{p^m}={(a-b)}^{p^m}\in F$

This suggests that $(a-b)$belongs to L i.e. $(a-b)\in L$,

Now, $a\in L$, $b\in L$

Then, $(a,b)\in K$such that $a^{p^m},b^{p^m}\in F$

Now, to prove that $\left(a{b}^{-1}\right)\in F$

As, $b^{p^m}\in F$

The above consideration suggests that ${\left(b^{p^m}\right)}^{-1}\in F$

Therefore,

$\left(a^{p^m}\right)\cdot {\left(b^{p^m}\right)}^{-1}\in F$

Which gives ${\left(a{b}^{-1}\right)}^{p^m}\in F$

Thus, $\left(a{b}^{-1}\right)\in F$

Nos, both the properties are satisfied.

Thus, L is a subfield of K that contains F.

Hence, proved.

Step 3:Prove that  (b)

To prove that: L=F

Define a map$g:K\to K$ such that $g\left(a\right)=a^{p^m}$

To prove that g is an isomorphism.

Proof:

Consider

$\begin{array}{c}f(a,b)={(a,b)}^{p^m}\\ =a^{p^m}{b}^{p^m}\\ =f\left(a\right)\cdot f\left(b\right)\end{array}b$

And, as

$\begin{array}{c}f(a+b)={(a+b)}^{p^m}\\ =a^{p^m}+b^{p^m}\\ =f\left(a\right)+f\left(b\right)\end{array}$

Therefore, f is a homomorphism since it is a map between two algebraic structures of the same type i.e, $f\left(a\right)$and $f\left(b\right)$.

Now, $g\left(F\right)=F$

Which suggests that $g^{-1}\left(f\right)=F$

Which gives

role="math" localid="1659341351012" $\begin{array}{c}{g}^{-1}\left(F\right)=\{a\in K|a^{p^m}\in F\}\\ =L\end{array}$

Thus, $L=F$

Hence, by using homomorphism, $L=F$ is proved.