Question

6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be a subgroup of G. 11. State Lagrange's Theorem. 12. Define the symmetric group S. True-False and Short Answer. 5 points each. (a) True or false: Given the positive integer n, Zn {0} is a group under multiplication. Justify your answer. (1 2 3 4 5 6 7 8 i (b) What power of the permutation a = (3 is the identity? (3 2 1 4 7 5 6 8) (c) Consider the elements o = (123)(124) and T = (1,3)(2, 4) in SCalculate o-173. (d) True or false: A finite group G of prime order has only {e} and Gas subgroups. (e) True or false: There exists an irreducible polynomial of degree four in R[x]. (f) True or false: Every field has infinitely many elements. Justify your answer. (g) Find the multiplicative inverse of 2 + 2a in the field Q(a) SC, where a is a root of x² + 3+ 4. (h) True or false: The ring Z12 is an integral domain. Justify your answer. (i) Give an example of a proper subgroup of GL,(R). 6) Give a concrete example of a group G, with subgroup H, and element ge H such that gH #Hg. Proof presentations. Provide complete proofs of the following statements. 15 points each. 1. The Rational Root Test. Lagrange's Theorem (including the preliminary steps leading up to the proof as presented in class
Q9

Answer 1

Fundamental Theorem of Algebra:-

Given any positive integer $n\geq 1$ and any choice of complex numbers such that $a_n \neq0,$ the polynomial equation

$a_nz^n+a_{n-1}z^{n-1}+...+a_1z+a_0=0$ has at least one solution $z\in C$ .