Subfields of an Infinite field of char p Do all infinite fields of char p contain a subfield isomorphic to $F_{p}(x)$?
 A: HINT $\ $ Consider the algebraic closure of $\rm\:\mathbb F_p\:,\:$ recalling that algebraically closed fields are infinite, since for any finite field $\rm\:\mathbb F\:$ with elements $\rm\:a_1,\cdots,a_n,\:$ a la Euclid, we can construct the polynomial $\rm\:1 + (x-a_1)\cdots(x-a_n)\:$ coprime to all of the primes $\rm\: x-a_i\:,\:$ hence having no roots $\rm\:a_{\:i}\:$ in $\rm\:\mathbb F\:.$ 
A: Perhaps this is an opportunity to walk through a mechanical process from the statement of the problem to its solution. Suppose $F$ is a field of characteristic $p$ which does not contain a subfield isomorphic to $\mathbb{F}_p(x)$. In other words, no element of $F$ is transcendental over $\mathbb{F}_p$. It follows that every element of $F$ must instead be algebraic over $\mathbb{F}_p$, hence that $F$ must be a subfield of $\overline{\mathbb{F}_p}$. As no finite field is algebraically closed, $\overline{\mathbb{F}_p}$ is an infinite field of characteristic $p$ which contains no transcendental elements, and we are done. 
