The following is a text that I'm not quite understanding
"The set of rational expressions $K(t)$ is a transcendental extension of the subfield $K$ of $ℂ$.
PROOF Clearly $K(t)$ is a simple expression generated by $t$..."
How is it that $K(t)$ is a simple expression generated by $t$? Simple extensions are formed by adjoining an element to a certain field, $t$ is a variable, not an element, so how could $K(t)$ possibly be a simple field extension?
"...If $p$ is a polynomial over $K$ such that $p(t)=0$, then $p=0$ by definition of $K(t)$, so the extension is transcendental."
Does the latter sentence has anything to do to the fact that the only polynomial $p$ that will always equal $0$ regardless of the variable we plug in is $p=0$? Is the only sense I can get out of what the author is saying. Is is that we are looking each polynomial in $K(t)$ as an element? If this is the case, then still, how is it that $K(t)$ is formed from adjoining $t$ to $K$?
I would really appreciate any help/thoughts.
$K(t)$ is the set of rational function ${{p(t)}\over {q(t)}}$ where $p,q$ are polynomials. It is a trancendantal extension since $t$ is not algebraic over $K$, that is you cannot find a polynomial $p$ such that $p(t)=0$