(a) State Brouwer's fixed point theorem in the plane and prove that the statement is equivalent to non-existence of a continuous retraction of the closed disk $D$ to its boundary $\partial D$.

(b) Use Brouwer's fixed point theorem to prove that there is a complex number $z$ in the closed unit disc such that $z^{6}-z^{5}+2 z^{2}+6 z+1=0$.