Homework Answers
Suppose a, b e Z. Show that if a for some integer c E Z, then...
Suppose a, b e Z. Show that if a for some integer c E Z, then a or b is even. Hint: Let P, A, and B be respectively the statements P {a2 + b2-c2}, A-fa is even), and B b is even]. In this problem you have to show that P (AVB). Use contradiction, i.e., prove that if the negation of this statement is true, then you come to a contradiction. Use that so that you have to assume...
Use mathematical induction to show that P(A1∩A2∩...∩An) = P(A1)P(A2|A1)P(A3|A1∩A2)....P(An|A1∩A2∩...∩ An-1) You can assume that you know...
Use mathematical induction to show that P(A1∩A2∩...∩An) = P(A1)P(A2|A1)P(A3|A1∩A2)....P(An|A1∩A2∩...∩ An-1) You can assume that you know P(A|B) = P(A|B)P(B)
4. The rational function 1 p(2) = (z – i)4 + 4 is holomorphic on the...
4. The rational function 1 p(2) = (z – i)4 + 4 is holomorphic on the domain C\{a1, A2, A3, A4} for some four distinct points d1, A2, A3, and 04. (a) Find the values of a1, A2, A3, and 04. [8] (b) Use one of the Cauchy's formulas to evaluate the integral of p(2) along y, a positively oriented closed contour parametrising a rect- angle with vertices Fi and 4£i. Show that this integral can be expressed in the...
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic re...
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic residues of p that lie between 1 and p - 1. Prove that 1,0 (P-1)/2 i- 1 Hint: If a is a quadratic residue less than or equal to (p-1)/2 then what is p - ai?
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are...
(a) Let G be a group of order 4 with identity e. Show that G is either cyclic or a2-e for all (b)...
(Abstract Algebra) Please answer a-d clearly. Show your work and
explain your answer.
(a) Let G be a group of order 4 with identity e. Show that G is either cyclic or a2-e for all (b) Does the result of part (a) generalize to groups of order p2 for any positive integer p? In other words, is it the case that if G is a group of order p2 with identity e, then is either cyclic or a- e for...
(1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1)) (b) Let φο : Zp[2] Zp be the evaluation homomorphism at 0. Compute φο(z...
(1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1)) (b) Let φο : Zp[2] Zp be the evaluation homomorphism at 0. Compute φο(zp-1-1) and φο((1-1)(1-2) . . . (z-(p-1))) (c) Use parts (a) and (b) to conclude that (-1)--1.
(1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1))...
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if...
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in
Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q
contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q)
is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP +
βQ|α, β ∈ Z[x]}.
(iii) For which primes p and which integers n ≥ 1 is the
polynomial xn − p...
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 +...
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)...
(b) Let p be a prime that is congruent to 3 modulo 4. Let b ∈...
(b) Let p be a prime that is congruent to 3 modulo 4. Let b ∈ Z. Let a = b (p+1)/4 . Show that a 2 ≡ ±b (mod p). (c) Give an algorithm to compute square roots of something modulo p, when p ≡ 3 (mod 4). Note: Not all things are square modulo p, so the algorithm should return the square root or inform you there is no square.
group theory Example 9.5 Show the funda mental group for 2-complex {e,f",efe e is isomorphic to Z, xZ, for p and...
group theory
Example 9.5 Show the funda mental group for 2-complex {e,f",efe e is isomorphic to Z, xZ, for p and q are relatively prime. Solution arch hp
Example 9.5 Show the funda mental group for 2-complex {e,f",efe e is isomorphic to Z, xZ, for p and q are relatively prime. Solution arch hp
Suppose a, b e Z. Show that if a for some integer c E Z, then a or b is even. Hint: Let P, A, and B be respectively the statements P {a2 + b2-c2}, A-fa is even), and B b is even]. In this problem you have to show that P (AVB). Use contradiction, i.e., prove that if the negation of this statement is true, then you come to a contradiction. Use that so that you have to assume...
4. The rational function 1 p(2) = (z – i)4 + 4 is holomorphic on the domain C\{a1, A2, A3, A4} for some four distinct points d1, A2, A3, and 04. (a) Find the values of a1, A2, A3, and 04. [8] (b) Use one of the Cauchy's formulas to evaluate the integral of p(2) along y, a positively oriented closed contour parametrising a rect- angle with vertices Fi and 4£i. Show that this integral can be expressed in the...
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic residues of p that lie between 1 and p - 1. Prove that 1,0 (P-1)/2 i- 1 Hint: If a is a quadratic residue less than or equal to (p-1)/2 then what is p - ai?
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are...
(Abstract Algebra) Please answer a-d clearly. Show your work and
explain your answer.
(a) Let G be a group of order 4 with identity e. Show that G is either cyclic or a2-e for all (b) Does the result of part (a) generalize to groups of order p2 for any positive integer p? In other words, is it the case that if G is a group of order p2 with identity e, then is either cyclic or a- e for...
(1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1)) (b) Let φο : Zp[2] Zp be the evaluation homomorphism at 0. Compute φο(zp-1-1) and φο((1-1)(1-2) . . . (z-(p-1))) (c) Use parts (a) and (b) to conclude that (-1)--1.
(1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1))...
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in
Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q
contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q)
is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP +
βQ|α, β ∈ Z[x]}.
(iii) For which primes p and which integers n ≥ 1 is the
polynomial xn − p...
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)...
group theory
Example 9.5 Show the funda mental group for 2-complex {e,f",efe e is isomorphic to Z, xZ, for p and q are relatively prime. Solution arch hp
Example 9.5 Show the funda mental group for 2-complex {e,f",efe e is isomorphic to Z, xZ, for p and q are relatively prime. Solution arch hp
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