Let R denote the ring of Gaussian integers, i.e., the set of all complex numbers a + bi with a, b ∈ Z. Define
N : R → Z by N(a + bi) = a^2 + b^2.
(i) For x,y ∈ R, prove that N(xy) = N(x)N(y).
(ii) Use part (i) to prove that 1, −1, i, −i are the only units in R.
Let R denote the ring of Gaussian integers, i.e., the set of all complex numbers a + bi with a, b...
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli 5. Let Zli_ {a + bi l a,b...
Definition:In the complex numbers, let J denote the set, {x+y√3i :x and y are in Z}. J is an integral domain containing Z. If a is in J, then N(a) is a non-negative member of Z. If a and b are in J and a|b in J, then N(a)|N(b) in Z. The units of J are 1, -1 Question:If a and b are in J and ab = 2, then prove one of a and b is a unit. Thus,...
Define where S is the collection of all real valued sequences i.e. S = {x : N → R} and we denote xi for the ith element a the sequence x E S. Take for any x EL (i) Show that lic 12 (where recall 1-(x є s i Izel < oo)) (ii) Is l? Prove this or find a counterexample to show that these two sets do not coinside (iii) ls e c loc where recall looー(x є sl...
5. Let R denote the set of real numbers. Which of the following subsets of R xR can be written as Ax B for appropriate subsets A, B of R? In case of a positive answer, specify the sets A and B. (a) {(z,y)12z<3, 1<y< 2}, (b) {z,)2+y= 1), (c) {(z,y)|z= 2, y R), (d) {(z,y)|z,yS 0}, (e) {(z,y) z y is an integer).
Theorem 16.1. Let p be a prime number. Suppose r is a Gaussian integer satisfying N(r) = p. Then r is irreducible in Z[i]. In particular, if a and b are integers such that a² +62 = p, then the Gaussian integers Ea – bi and £b£ai are irreducible. Exercise 16.1. Prove Theorem 16.1. (Hint: For the first part, suppose st is a factorization of r. You must show that this factorization is trivial. Apply the norm to obtain p=...
2.) Let Z the set of integers and two binary operations on it: Z23(x,y) → xTy = xy + 3x +3y +6 e Z i) Show (Z,L,T)is an integral domain ii) Find the set of units U(Z) 2.) Let Z the set of integers and two binary operations on it: Z23(x,y) → xTy = xy + 3x +3y +6 e Z i) Show (Z,L,T)is an integral domain ii) Find the set of units U(Z)
1. Recall the definition of red, green, blue numbers. Let R denote the set of red numbers. Let G be the set of green numbers, and let B denote the set of blue numbers. Is R S G S B = Z. Here Z is the set of all intergers. Explain.
(a) Let R be a commutative ring. Given a finite subset {ai, a2, , an} of R, con- sider the set {rial + r202 + . . . + rnan I ri, r2, . . . , rn є R), which we denote by 〈a1, a2 , . . . , Prove that 〈a1, a2, . . . , an〉 įs an ideal of R. (If an ideal 1 = 〈a1, аг, . . . , an) for some a,...
(6 pts) Alternate construction of the integers from the natural numbers. Suppose that the natural numbers N = {0,1,2, ...} ations. We do not yet have a notion of subtraction or the cancellation law for addition (if x+y = x+ z, then y = 2) and for multiplication given with the usual addition and multiplication oper negative numbers, though we do have are Define a relation R on N2 as follows (a, b) R (c, d) if and only if...
Abstract Algebra Ring Question. see the image and show parts a, b, c, and d please. 36. Let R be a ring with identity. (a) Let u be a unit in R. Define a map ix : R R by Huru". Prove that i, is an automorphism of R. Such an automorphism of R is called an inner automorphism of R. Denote the set of all inner automorphisms of R by Inn(R). (b) Denote the set of all automorphisms of...