Sums/Products of Algebraic Numbers are also Algebraic, a Constructive Proof

Anamay Tengse
Gunjan Kumar
Friday, 9 Aug 2019, 17:15 to 18:45
A-201 (STCS Seminar Room)
Abstract: A complex number z is said to be algebraic, if there is a univariate f(x) with real coefficients such that f(z)=0. For instance i, the square root of -1, is algebraic with f(x) being x^2 + 1.
Now given that z_1 and z_2 are algebraic, suppose you want to show that (z_1 + z_2) or (z_1 * z_2) are also algebraic. In other words, given polynomials f(x) and g(x) with z_1 and z_2 as (one of their) roots, we want to construct polynomials that have (z_1 + z_2) or (z_1 * z_2) as a root. In this talk we will build such polynomials via an interesting object called the resultant.
P.S.: Little background will be assumed, so if the problem statement is clear then so should be the talk.