September 8th
Today I learned the definition of perfect fields, from Keith Conrad : $K$ is perfect if and only if all irreducible polynomials $K[x]$ are separable. (Separable means that it doesn't have double roots when factored in the algebraic closure.) So $\QQ$ is perfect because for $\pi(x)\in\QQ[x],$ we can look at $\gcd(\pi(x),\pi'(x)),$ which will have degree less than $\pi(x)$ and divide into $\QQ[x]$ but divide $\pi(x).$ If $\pi(x)$ is linear, then we're already done, but if $\deg\pi \gt 1,$ then $\deg\pi'=\deg\pi-1 \gt 0,$ so\[\deg\gcd(\pi,\pi') \gt 0,\]which is our contradiction to the irreducibility of $\pi.$ The necessary hypothesis we used was that $\QQ[x]$ has characteristic $0$ so that $\deg\pi'=\deg\pi-1$ by direct computation.