Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 3614. Let α be a zero of x 3 + x 2 + 1 over Z 2 . Construct a finite field of order 8.Show that x 3 + x 2 + 1 splits in Z 2 (α).5. Construct a finite field of order 27.6. Prove or disprove: Q ∗ is cyclic.7. Factor each of the following polynomials in Z 2 [x].(a) x 5 − 1(c) x 9 − 1(b) x 6 + x 5 + x 4 + x 3 + x 2 + x + 1(d) x 4 + x 3 + x 2 + x + 18. Prove or disprove: Z 2 [x]/〈x 3 + x + 1〉 ∼ = Z 2 [x]/〈x 3 + x 2 + 1〉.9. Determine the number of cyclic codes of length n for n = 6, 7, 8, 10.10. Prove that the ideal 〈t + 1〉 in R n is the code in Z n 2 consisting of all words ofeven parity.11. Construct all BCH codes of(a) length 7.(b) length 15.12. Prove or disprove: There exists a finite field that is algebraically closed.13. Let p be prime. Prove that the field of rational functions Z p (x) is an infinitefield of characteristic p.14. Let D be an integral domain of characteristic p. Prove that (a − b) pn =a pn − b pn for all a, b ∈ D.15. Show that every element in a finite field can be written as the sum of twosquares.16. Let E and F be subfields of a finite field K. If E is isomorphic to F , showthat E = F .17. Let F ⊂ E ⊂ K be fields. If K is separable over F , show that K is alsoseparable over E.18. Let E be an extension of a finite field F , where F has q elements. Let α ∈ Ebe algebraic over F of degree n. Prove that F (α) has q n elements.19. Show that every finite extension of a finite field F is simple; that is, if E isa finite extension of a finite field F , prove that there exists an α ∈ E suchthat E = F (α).20. Show that for every n there exists an irreducible polynomial of degree nin Z p [x].21. Prove that the Frobenius map φ : GF(p n ) → GF(p n ) given by φ : α ↦→ α pis an automorphism of order n.