27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × Z/nZ whith f(x) = (x + mZ, x + nZ) is surjective (b) Prove the Chinese Remainder Theorem: If m and n are relatively prime integers > 1 and if a and b are any integers, then there exists a E Z such that b(mod n). a(mod m) and a a Hint: (a)] 27. (a) Let...
Let n be a nonnegative integer and let F 22 + 1 be a Fermat number. Prove that if is a prime number, then either n=0 or 3--1mod F. [Hint: If n 2 1, use the law of quadratic reciprocity to evaluate the Legendre symbol (3/F). Now use Euler's Criterion (Theorem 4.4).] Let n be a nonnegative integer and let F 22 + 1 be a Fermat number. Prove that if is a prime number, then either n=0 or 3--1mod...
mk-()s (m2'). m+1 [k] be a surjective map. Show that Σ'ıf(j) 2. Let 1 kS m and let f : [m] mk-()s (m2'). m+1 [k] be a surjective map. Show that Σ'ıf(j) 2. Let 1 kS m and let f : [m]
5. Let CONTAINPDA DFA L(M1) C (M2)}. Show that CONTAIN PDA DFA is decidable. {{M1, M2) M1 is a PDA and M2 is a DFA such that = 5. Let CONTAINPDA DFA L(M1) C (M2)}. Show that CONTAIN PDA DFA is decidable. {{M1, M2) M1 is a PDA and M2 is a DFA such that =
Let F be a field of characteristic p and suppose that F ⊂ L is separable and that p | [L : F]. Suppose furthermore that any q-th root of unity, where q is prime and q ≡ 1 (mod p), that lies in L already lies in F. Show that F ⊂ L cannot be solvable
Let pEN be a prime and let V be a vector space over Zp with dimension n. Show that |v| = pn. Let pEN be a prime and let V be a vector space over Zp with dimension n. Show that |v| = pn.
a set function, λ on S by λ((a, b) F(b)--F(a) and λ(0) 1. Show that if Eİ, E2 E S then Ei n E2 ES and Ei ~ E2 is a finite disjoint union of 0. sets in S 2. Show that the o-algebra generated by S is the Borel o-algebra on R. 3. Show that if E and Ea are disjoint sets in S and A U S, then (A) A(E)+A(B2). 4, Show that if E. .. ova natn...
Rings and fields- Abstract Algebra 2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g() e Fx be any polynomial. Show that every irreducible factor of f(g()) E Flx] has degree divisible by n (b) (4 points) Prove that Q(2) is not a subfield of any cyclotomic field over Q. 2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g()...
5. (10 marks) (a) Let E, E2} be mutually independent random variables. Show that the conditional density T(e1, e2 x) can be written in the form (4 marks) (b) Let a set of observations Y be of the form Yk exp(r)Ek , k = 1,... , M where ER.Let Ek be mutually independent and identically distributed and normal with T(ek) N(He,©?) for all k (i) Derive the likelihood density n(y|x) (ii) Derive the maximum likelihood estimate ML (3 marks) (3...
= Let R be a ring (not necessarily commutative) and let I be a two-sided ideal in R. Let 0 : R + R/I denote the natural projection homomorphism, and write ř = º(r) = r +I. (a) Show that the function Ø : Mn(R) + Mn(R/I) M = (mij) Ø(M)= M is a surjective ring homomorphism with ker ý = Mn(I). (b) Use Homework 11, Problem 2, to argue that M2(2Z) is a maximal ideal in M2(Z). (c) Show...