Question

Question on #2 parts a and b (second photo has definitions).

2a. Write out A3+3 explicitly as a polynomial in r, s ,t 13 + u as a polynomial in the three elementary symmetric funetions in r, s, t the matrix 82 Let σ be a permutation in Sym({r,s,t) Lemma 10.1. then If σ has order 2, then σ interchanges A3 and μ that every permutation in Sym(ir,,t) fixes both 3,pe, every In any case, it follows Аз + 3 and in r, s,t. As a somewhat tedious exercise, you are symmettioxplicitly as a polynomial in r,s,t, and then to will be asked to write out A3 + μ3 explicitly as a D lynewhat express it as a polynomial three elementary Note that, in this case, ll be asked to momial in the three elementary symmetric functions in r,s.t. (2) 82-rs +rt +st p, and (3) 83 = rst= q. Now A3 and μ3 are the two roots of the quadratic polynomial Hence, using the Quadratic Formula, we could explicitly solve for A3 and u? in terms of p and q. Then, by taking cube roots, we could find A and u. Finally, we end up with a system of three linear equations in the three unknowns r, s, and t: r+s+t 0 to be solved in order to find r, s, and t. We leave as an exercise for you to verify that the coefficient matrix is invertible, and hence the system has a unique solution Lagrange further extended these ideas to explain the solution of thequartie equation. We give a brief description in the general spirit of his work. C quartic Let the roots of f(x) be ri, /3. 3, and ra. Consider the (ri + r2)(r3 + r4) (ni + r3)(r2 +r) 02

Answer 1

Solutfons b. Expoess λ, 3 as a Payonfal fn the three elementany c onctfons. n b Consides Prom e2O,