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Question 2. In this exercise, you will show that Z[V-5] is not a U.F.D. (but it...
5. A generalization of the polynomial ring Z[2] is the ring Z[[2]] of formal power series...
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...
2. (The Ring Z[V-5]) We saw that Z[V-5 doesn't have a Nevertheless, exercise hints at some...
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)...
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modu...
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modulo p(x) and Fs denote the field Zs[r] modulo p(x). (i) Prove that p(x) is irreducible over Q and also irreducible over Zs, so that in fact, F and Fs are fields (ii) Calculate 1+2r2-2r + in HF. (iii) Find the multiplicative inverse of 1 +2r2 in F. (iv) Repeat (ii) and (iii) for Fs. (v) How many elements are in...
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a +...
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a + 1 a) (5 points) Show that the quotient rga)/ () is a field. b) (5 points) Let a denote the coset z()Regarding F as a vector space over Z2, find a basis for F coasisting of powers of a c) (5 poluts) How nuany elements dors F have? Justify your answer. d) (5 points) Compute the product afas t a) i.e. expand this product...
2. (5 points) Let fi)+z and gla) r+1 Find if o gz) and expand your answer...
2. (5 points) Let fi)+z and gla) r+1 Find if o gz) and expand your answer 3 (6 polats) Lit le)-+5 and te)Show that f and g are inverses of each other 2, show that f and g are inverses of each other -
8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random...
8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random variables with mean u < o0. Let Z, = ... Хо. X.
8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random variables with mean u
2. Fix m, n E N. Define a mapping f:Z/nZ+Z/mZ by f([a]n) = [a]m. a. Prove...
2. Fix m, n E N. Define a mapping f:Z/nZ+Z/mZ by f([a]n) = [a]m. a. Prove that if m | n then f is a well-defined function. That is, prove that if (a)n = [b]n then f([a]n) = f([b]n). b. Let n = 12 and m = 3. Write PreImp({[1]3, [2]3}) in roster notation. c. Suppose mfn. Show that f is ill-defined. That is, show there exist a, b E Z such that (a)n = [b]n but f([a]n) + f([b]n).
Exercises Exercise 4.1. Lete: G → GL(U), ψ: G → GL(V) and : representations of a group G. Suppose...
Please Complete 4.1.
Exercises Exercise 4.1. Lete: G → GL(U), ψ: G → GL(V) and : representations of a group G. Suppose that Te HomG(φ, ψ) and Se Prove that ST Homc(p.,p). p: G GL(U Xp. Prove tha Exercise 4.2. Let o be a representation of a group G with character Exercise 4.3. Let p: GGL(V) be an irreducible representation Let be the center of G. Show that if a e Z(G), then p(a) Exercise 4.4. Let G be a...
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is uni...
Number Theory
13 and 14 please!
13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
Question 2 please Exercise 1. Define an operation on Z by a b= a - b....
Question 2 please
Exercise 1. Define an operation on Z by a b= a - b. Determine ife is associative or commutative. Find a right identity. Is there a left identity? What about inverses? Exercise 2. Write a multiplication table for the set A = {a,b,c,d,e} such that e is an identity element, the product is defined for all elements and each element has an inverse, but the product is NOT associative. Show by example that it is not associative....
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...
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)...
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modulo p(x) and Fs denote the field Zs[r] modulo p(x). (i) Prove that p(x) is irreducible over Q and also irreducible over Zs, so that in fact, F and Fs are fields (ii) Calculate 1+2r2-2r + in HF. (iii) Find the multiplicative inverse of 1 +2r2 in F. (iv) Repeat (ii) and (iii) for Fs. (v) How many elements are in...
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a + 1 a) (5 points) Show that the quotient rga)/ () is a field. b) (5 points) Let a denote the coset z()Regarding F as a vector space over Z2, find a basis for F coasisting of powers of a c) (5 poluts) How nuany elements dors F have? Justify your answer. d) (5 points) Compute the product afas t a) i.e. expand this product...
2. (5 points) Let fi)+z and gla) r+1 Find if o gz) and expand your answer 3 (6 polats) Lit le)-+5 and te)Show that f and g are inverses of each other 2, show that f and g are inverses of each other -
8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random variables with mean u < o0. Let Z, = ... Хо. X.
8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random variables with mean u
2. Fix m, n E N. Define a mapping f:Z/nZ+Z/mZ by f([a]n) = [a]m. a. Prove that if m | n then f is a well-defined function. That is, prove that if (a)n = [b]n then f([a]n) = f([b]n). b. Let n = 12 and m = 3. Write PreImp({[1]3, [2]3}) in roster notation. c. Suppose mfn. Show that f is ill-defined. That is, show there exist a, b E Z such that (a)n = [b]n but f([a]n) + f([b]n).
Please Complete 4.1.
Exercises Exercise 4.1. Lete: G → GL(U), ψ: G → GL(V) and : representations of a group G. Suppose that Te HomG(φ, ψ) and Se Prove that ST Homc(p.,p). p: G GL(U Xp. Prove tha Exercise 4.2. Let o be a representation of a group G with character Exercise 4.3. Let p: GGL(V) be an irreducible representation Let be the center of G. Show that if a e Z(G), then p(a) Exercise 4.4. Let G be a...
Number Theory
13 and 14 please!
13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
Question 2 please
Exercise 1. Define an operation on Z by a b= a - b. Determine ife is associative or commutative. Find a right identity. Is there a left identity? What about inverses? Exercise 2. Write a multiplication table for the set A = {a,b,c,d,e} such that e is an identity element, the product is defined for all elements and each element has an inverse, but the product is NOT associative. Show by example that it is not associative....
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