Discriminants and Galois Groups

Compute the discriminants and the Galois groups of the polynomials x3 + 27x − 4 x4 − 5

Symmetric Functions

Please see the attached file for the fully formatted problems. Write the following functions in terms of the elementary symmetric functions: (u1 + u2)(u1 + u3)(u3 + u2) u31 + u32 + u33 + u34

Ring Isomorphism

Please see the attached file for the fully formatted problem. Let gcd(m, n) = 1. We know that Zmn = Zm × Zn as an additive group. Is there a ring isomorphism Zmn = Zm × Zn? Same question for the complex numbers C with C = R × R as an additive group.

Galois Groups of Polynomials

Please see the attached file for the fully formatted problems. Describe the Galois groups of the polynomials x3 + 27x − 4 x4 − 5

Irreduible Polynomial : Splitting Field

Let K be obtained as a field Q(alpha) where alpha is a root of P(x) = x3 −3. Find an irreducible polynomial which defines the splitting field of P(x).

Finite Fields : Field Extensions

Please see the attached file for the fully formatted problems. Show the existence of an extension of Fq of order l for any prime l.

Groups of Order 56

Please see the attached PDF file. I would prefer the solution as a PDF file. Thanks!

Quotient Ring

Please see the attached PDF file. I would prefer a solution in PDF format. Thanks!

Irreducible Polynomial over a Field

Please see the attached file for the fully formatted problems. 5. Find an irreducible polynomial f(x) over the field Z3 with Z3[x]/(f(x)) = F243. Note that 243 = 3^5 . Please explain your reasoning and solution in as much detail as possible. Thank You.

Groups of Order 56 : 13 Isomorphisms

Groups of Order 56. The problem is out of Dummit and Foote in the section on Semidirect Products, which says that there are 13 isomorphism types for such a group.