
Question 15 6 pts The polynomial g(x) = 2 + x + x?is irreducible over Z3,...
6 pts Question 16 The polynomial g(x) = 2 + x + x?is irreducible over Zz, and B is a root of g(x). What is an equivalent expression for 1 + 87 as a power of B? O B3 O 06
Exercise 2 (pts 5). Let g() E Z[2]. Prove that g(x) is irreducible over Zx if and only if g() is irreducible as polynomial in Q[o].
Rings and fields- Abstract Algebra
2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g() e Fx be any polynomial. Show that every irreducible factor of f(g()) E Flx] has degree divisible by n (b) (4 points) Prove that Q(2) is not a subfield of any cyclotomic field over Q.
2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g()...
Question 14 5pt In the options below, there is only one polynomial that is irreducible over the field Z2. Determine that polynomial. 1 + x2 1+ x + 25 O x + x² + x3 0 1 + x + x2 + 23 01+ x + x2 + x3 + x4 0 1 + x3
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
4. Show that the polynomial g(x) = x++x+1 is irreducible over Z2. In the quotient ring Z2[x]/(g(x)) let S = x+(g(x)), so that Z2[x]/(g(x)) = Z2(). Express 85 and (82 +1)-1 in the form a + b8 + 082 +883, where a, b, c, d e Z2.
Polynomial over the Fields: a) If p(x) an element of F[x] is not irreducible, then there are at least two polynomials g(x) and h(x), neither which is a constant, such that p(x)=g(x)h(x). Explain b) Use problem a to prove: If p(x) is not irreducible, then p(x)=j(x)k(x), where both j(x) and k(x) are polynomials of lower degree than p(x).
Let F=Z_3, the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (ab)^3=(a^3)(b^3) The Remainder Theorem If a and b are in Z_3 then (a+b)^3=2^3+b^3 For all a in Z_3, a^3=a The first isomorphism...
6. One root of the polynomial f(x) = 2x5 – 23x4 + 76x3 – 9x2 – 246c +234 over C is 5 - i. (a) Write f(x) as a product of irreducible polynomials in Q[x]. Show your work. (b) Write f(x) as a product of irreducible polynomials in R[x]. Show your work. (c) Write f(x) as a product of irreducible polynomials in C[x]. Show your work.
Consider the polynomial f(x) = x p − x + 1 ∈ Zp[x]. (a) Let a be a root of f in some extension. Prove that a /∈ Zp and a + b is a root of f for all b ∈ Zp. (b) Prove that f is irreducible over Zp. [Hint: Assume it is reducible. If one of the factors has degree m, look at the coefficient of x m−1 and get a contradiction.]