In this post we show the Mason-Stothers theorem, the so-called $abc$ theorem for polynomials, and derive Fermat's Last theorem and Davenport's inequality for polynomials. These three theorems correspond to the $abc$ conjecture, Fermat's Last Theorem and Hall's conjecture in number theory.
We compute the analytic continuation of the Riemann Zeta function and after that the reader will realise that asserting $1+2+\dots=-\frac{1}{12}$ without enough caution is not a good idea.
In this post we study cyclotomic polynomials in field theory and deduce some baisc properties of it. We will also use it to solve some problems in field theory.
We prove the celebrated Hensel's lemma using the so-called Newton's method and "double induction", and try to find solutions of polynomials in $\mathbb{Q}_p$.