Question

If K is a field of prime characteristic p prove that its prime subfield is the intersection of all the subfield of K.

Short answer

Prime subfield of K is the intersection of all the subfield of K.

Step-by-step solution

Definition of subfield.

A subfield of a mathematical field that is itself a field that mean a subset of a given field which is a field in itself is called as a subfield.

Showing that prime subfield of K is the intersection of all subfield of K

Let K be a field of prime characteristic p . It is enough to show that prime field of Kis the smallest subfield of K.

Therefore it is needed to show that if K has prime characteristic p then prime field of K is isomorphic to F_p and representation of F_p is $\left\{0,1,2...p-1\right\}$.

Define a mapping $\varphi :F_p\to K$as:

$$\begin{gathered} r=r.1 \\ =1+1+1.....+1\left(rtimes\right) \end{gathered}$$

Here 1 is the multiplication unit in K denotes addition in K. As $\varphi$is additive and multiplicative this is a field homorphis. To show that $F_p$is embedded in K it is remained to show that the mapping is injective.

Assume on the countrary for somelocalid="1659091723355" $p>r>s\ge 0$:

$\varphi \left(r\right)=\varphi \left(s\right)$

Put $c=r-s>0$ by definition of r and s,c is an element of F_p . Thus it has a multiplication inverse $c^{-1}$ in F_p. Therefore:

$\begin{array}{rcl}\mathrm{\varphi }\left(1\right)& =& \mathrm{\varphi }\left({\mathrm{cc}}^{-1}\right)\\ & =& \mathrm{\varphi }\left(\mathrm{c}\right)\mathrm{\varphi }\left({\mathrm{c}}^{-1}\right)\\ & =& \mathrm{\varphi }\left(\mathrm{r}-\mathrm{s}\right)\mathrm{\varphi }\left({\mathrm{c}}^{-1}\right)\end{array}$

By definition of $\varphi$:

$\begin{array}{rcl}\varphi \left(1\right)& =& 1\\ & \ne & 0\end{array}$

Since K is field because of this contradiction $\varphi$ is an isomorphism between F_p and image of homomorphism $\ln \left(\varphi \right)$.

This isomorphism proves that $\ln \left(\varphi \right)$ is a subfield of K. Since F_p has no nontrivial subfield. Therefore it is its own prime subfield. So, $\ln \left(\varphi \right)$is a prime field of $\ln \left(\varphi \right)$of K is isomorphic to F_p .