Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 27319. Cyclotomic Polynomials. The polynomialΦ n (x) = xn − 1x − 1 = xn−1 + x n−2 + · · · + x + 1is called the cyclotomic polynomial. Show that Φ p (x) is irreducible overQ for any prime p.20. If F is a field, show that there are infinitely many irreducible polynomials inF [x].21. Let R be a commutative ring with identity. Prove that multiplication iscommutative in R[x].22. Let R be a commutative ring with identity. Prove that multiplication isdistributive in R[x].23. Show that x p − x has p distinct zeros in Z p [x], for any prime p. Concludethat thereforex p − x = x(x − 1)(x − 2) · · · (x − (p − 1)).24. Let F be a ring and f(x) = a 0 + a 1 x + · · · + a n x n be in F [x]. Definef ′ (x) = a 1 + 2a 2 x + · · · + na n x n−1 to be the derivative of f(x).(a) Prove that(f + g) ′ (x) = f ′ (x) + g ′ (x).Conclude that we can define a homomorphism of abelian groups D :F [x] → F [x] by (D(f(x)) = f ′ (x).(b) Calculate the kernel of D if charF = 0.(c) Calculate the kernel of D if charF = p.(d) Prove that(fg) ′ (x) = f ′ (x)g(x) + f(x)g ′ (x).(e) Suppose that we can factor a polynomial f(x) ∈ F [x] into linear factors,sayf(x) = a(x − a 1 )(x − a 2 ) · · · (x − a n ).Prove that f(x) has no repeated factors if and only if f(x) and f ′ (x)are relatively prime.25. Let F be a field. Show that F [x] is never a field.26. Let R be an integral domain. Prove that R[x 1 , . . . , x n ] is an integral domain.27. Let R be a commutative ring with identity. Show that R[x] has a subring R ′isomorphic to R.28. Let p(x) and q(x) be polynomials in R[x], where R is a commutative ringwith identity. Prove that deg(p(x) + q(x)) ≤ max(deg p(x), deg q(x)).