3. Let a, b, c E Z such that ca and (a,b) = 1. Show that (c, b) = 1. 4. Suppose a, b, c, d, e E Z such that e (a - b) and e| (c,d). Show that e (ad — bc). 5. Fix a, b E Z. Consider the statements P: (a, b) = 1, and Q: there exists x, y E Z so that ax + by = 1. Bézout’s lemma states that: if P, then...
(3.) (a) Suppose that y: R S is a ring homomorphism. Please prove that (-a) = -f(a) for all a ER (b) Suppose R and S are rings. Define the zero function y: R S by pa) = Os for all GER. Is y a ring homomorphism? Please explain. (4.) Suppose that p is a prime number and 4: Z, Z, is defined by wa) = a.
Please solve all questions 1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
Please write carefully! I just need part a and c done. Thank you. Will rate. 3 This problem is to prove the following in the precise fashion described in class: Let O C R2 be open and let f: 0+ R have continuous partial derivatives of order three. If (ro, o) O a local maximum value at (To, Va) (that is, there exist r > 0 such that B. (reo) O and (a) Multivariable Taylor Polynomial: Suppose that f has...
3. Let t be the co-ordinate on A (C) and let z, y be the co-ordinates on A2(C). Let f 4z? + 6xy + x-2y® E C[x, y] and let C be the curve C-V((f)) C A2(C) (You may assume without proof that f is an irreducible polynomial, therefore C is irreducible and I(C)- (f).) (a) Show that yo(t) = (2t3, 2t2 + t) defines a morphism p : A1 (C) → C. [3 marks] (b) Show that (z. У)...
3. This problem is to prove the following in the precise fashion described in class: Let o sR be open and let f :o, R have continuous partial derivatives of order three. If (o, 3o) ▽f(zo. ) = (0,0),Jar( , ) < 0, and fzz(z ,m)f (zo,yo) -(fe (a ,yo)) a local maximum value at (zo, yo) (that is, there exists r 0 such that B,(zo, yo) S O and f(a, y) 3 f(zo, yo) for all (x, y) e...
5. For z, w E C, show the following identities. (a) z + w = z + W (b) zw = zw (c) |zw| = |2||w (d) () = 1 where w #0 (e) [2"| = |z|" where n is a positive or negative integer
(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...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...