Prove that Z[V-5] is a ring
2. (The Ring Z[V-5]) We saw that Z[V-5 doesn't have a Nevertheless, exercise hints at some patterns that persist. unique factorization theorem we can still do some interesting number theory here as the following (a) Find all primes p < 100 that can be written in the form a25b2 or 2a22ab+3b2 for some a, b e Z. (You might make a table for |a|, |b| small and then find the primes in them.) Separate them into two lists (F) (b)...
= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer (b) (5 Pts) Prove that o is surjective onto its image. Answer = a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer (b) (5 Pts) Prove that o is surjective onto its image. Answer
5. Let I be an ideal in a ring R. Prove that the natural ring homomorphism T: RRI has kernel equal to I.
Q 5. Let F be a field and consider the polynomial ring l (a) State the Division Algorithm for polynomials in Plrl. b) Let a e F. Prove that -a divides f(x) in Fix] if and only if (a)- (c) Prove that z-37 divides 42-1 in F43[z]. Q 5. Let F be a field and consider the polynomial ring l (a) State the Division Algorithm for polynomials in Plrl. b) Let a e F. Prove that -a divides f(x) in...
8. let salle &]: xy, 2 e R} a). Prove that (5, +,-) is a ring, where t' and are the usual addition and multiplication of matrices. (You may assume standard properities of matrix Operations ) b). Let T be the set of matrices in 5 of the form { x so]. Prove that I is an ideal in the ring s. c). Let & be the function f: 5-71R, given by f[ 8 ] = 2 i prove that...
Consider the ring Z[i] = {a + bil ab € Z} where i is the imaginary unit satisfying 12 = -1. (a) True or False? The principal ideal (2) is a prime ideal of Z[i). Prove or provide a counterexample. (b) Prove that (2) is not a maximal ideal of Z[i].
5. A generalization of the polynomial ring Z[2] is the ring Z[[2]] of formal power series over Z. Its elements are power series in < (i.e., polynomials of possibly "infinite degree"). See Example 25.8. The polynomial x + 1 is not a unit in Z[c]: there is no polynomial p E Z[<] such that (x + 1)p(x) = 1. Show that 2+1 is a unit in Z[[x]]. Hint: Clearly, the multiplicative inverse of 2 + 1 in the field of...
(4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine how many elements Zu/5+5i) has. (5) Let m,n be integers with m|n. Prove that the surjective ring homomor- phism Z/n -> Z/m induces a group homomorphism on units, and that this group homomorphism is also surjective. (4) Let p Z be a prime. Prove that zli/(p+1) has exactly ] p2 +1 elements. Use that 5+5i (2+i)(3+i) to determine...
Question 2. In this exercise, you will show that Z[V-5] is not a U.F.D. (but it is an I.D., as you proved last lecture!) You will learn a common trick for reasoning about irreducibility and primality in a ring - with the help of special multiplicative functions to Z>. (i) First, calculate the units in Z[V-5] [Hint: calculate inverses first, assuming you can divide ("work- ing in Q[V-5]", and then see which ones actually lie in Z[V-5]] (ii) Next, we...
5. Show that the ring Z[7] = {a+bV7 | a, b e Z} is an integral domain.