I am going through Morandi's Field and Galois Theory, and I am looking at Example $1.5$, Chapter I. It says (more or less):
If $k$ is a field, let $K=k(t)$ be the field of rational functions in $t$ over $k$. Let $f$ be a nonzero element of $K$ and let $F=k(f)$ be the set of all rational functions in $f$. If $f(t)=t^2$, then $K/F$ is an extension of degree $2$, with basis $\lbrace 1,t\rbrace$.
I am sure there is something silly that I do not see, but could someone explain why, in any field, a quotient of polynomials can be written as the sum of a quotient of polynomials whose terms all have even degree plus such a sum multiplied by $t$?
1) $a_0+a_1t+\cdots+a_nt^n=(a_0+a_2t^2+\cdots)+t(a_1+a_3t^2+\cdots)$, where $a_i\in k$. (For short, any polynomial $f(t)$ can be uniquely written as $g(t^2)+th(t^2)$.)
2) $\dfrac{a+tb}{c+td}=\dfrac{(a+tb)(c-td)}{c^2-t^2d^2}=\dfrac{ac-t^2bd}{c^2-t^2d^2}+t\dfrac{bc-ad}{c^2-t^2d^2}$, where $a,b,c,d\in k[t^2]$. (Here $k[t^2]$ denotes the subring of $k[t]$ consisting of polynomials whose all monomials have even degree.)
Alternative approach: the minimal polynomial of $t$ over $k(t^2)$ is $X^2-t^2$.