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9. For F as in 8, define N:F-Q by N(a+bv2)--22 (i) Prove that N(a3)-N(a)N(8), for all α, β E F. (...
10. Use 9 above to prove that the equation x^2 − 2y^2 = 1 has infinitely many solutions over Q. W...
10. Use 9 above to prove that the equation x^2 − 2y^2 = 1 has infinitely many solutions over Q. What can you conclude about the number of solutions over Z? (question9: For F as in 8, define N : F → Q by N(a + b√2) = a^2 − 2b^2. (i) Prove that N(αβ) = N(α)N(β), for all α,β ∈ F. (ii) Find an element u ∈ F such that N(u) = 1 and such that all of the...
2. Let K F be fields and a, β E F. Define K(a, β) to be the intersection of all subfields of F co...
2. Let K F be fields and a, β E F. Define K(a, β) to be the intersection of all subfields of F containing K, a B. Define K(a β) to be the set of all elements in F of the form pa, β)-g(a, β) 1, where P(z,y), g(z, y) are polynomials in Kr, y] and g(a,B)0 (i) Prove that K(a, B) are K(a, B) are subfields of F and that they are, in fact, the same subfield. (ii) Find...
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3]...
Advanced Calculus
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion.
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....
4.5 Define Try/F, (a) a +a ++a for any a E F. The element TrF/F, (a) is called the trace of a with respect to the ex...
4.5 Define Try/F, (a) a +a ++a for any a E F. The element TrF/F, (a) is called the trace of a with respect to the extension F4/F. (i) Show that Trm/F, (a) is an element of F4 for all a e F (ii) Show that the map TrF F, (a) Tr/F, F F4 is an F-linear transformation, where both Fm and F are viewed as Fg vector spaces over (iii) Show that Tr/F, is surjective (iv) Let BE F....
8. Define (n) to be the number of positive integers less than n and n. That...
8. Define (n) to be the number of positive integers less than n and n. That is, (n) = {x e Z; 1 < x< n and gcd(x, n) = 1}|. Notice that U (n) |= ¢(n). For example U( 10) = {1, 3,7, 9} and therefore (10)= 4. It is well known that (n) is multiplicative. That is, if m, n are (mn) (m)¢(n). In general, (p") p" -p Also it's well known that there are relatively prime, then...
Let Y_1~Gamma(α=3,β=3), Y_2~Gamma(α=5,β=1), and W=2Y_1+6Y_2. a) (9 pts) Find the moment generating function ofW Justify all...
Let Y_1~Gamma(α=3,β=3), Y_2~Gamma(α=5,β=1), and W=2Y_1+6Y_2.
a) (9 pts) Find the moment generating function ofW Justify all steps b) (3 pts) Based on your result in part (a), what is the distribution of W(name and parameters)? n 2N(O, I) 2. IfZ NO, 1), then Ux(1) 3. ItY Gmmaa,B) and W then Wx(n) - s, and i-1 7. y's~ Poisson(W (i-l, ,Rind) and U-ŽYi, then U-Poisson(XA) 8 If%-Gamma(a, β) (i-I, ,Rind) and U-ΣΥί , then U~Gamma( ,4 β).(Note: all same β) 9...
9. Let E be an extension field of a field F. (1) What does it mean...
9. Let E be an extension field of a field F. (1) What does it mean for an element z EE being algebraic over F? (2) What does it mean for an element z E E being transcendental over F? (3) Can you find any element r e C such that r is transcendental over Q? (4) Can you prove that if a E E is algebraic over F then (F(a): F] is finite? (5) Can you prove that if...
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...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the field Q. Consider z E Q(w). Find a, b, c, d e Q us + w-2 such that : a + bu + cu2 + du 9. Let E be an extension field of a field F. (1) What does it mean for an element z E E being algebraic over F? (2) What does it mean for an element z EF being transcendental...
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).
2. Let K F be fields and a, β E F. Define K(a, β) to be the intersection of all subfields of F containing K, a B. Define K(a β) to be the set of all elements in F of the form pa, β)-g(a, β) 1, where P(z,y), g(z, y) are polynomials in Kr, y] and g(a,B)0 (i) Prove that K(a, B) are K(a, B) are subfields of F and that they are, in fact, the same subfield. (ii) Find...
Advanced Calculus
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion.
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....
4.5 Define Try/F, (a) a +a ++a for any a E F. The element TrF/F, (a) is called the trace of a with respect to the extension F4/F. (i) Show that Trm/F, (a) is an element of F4 for all a e F (ii) Show that the map TrF F, (a) Tr/F, F F4 is an F-linear transformation, where both Fm and F are viewed as Fg vector spaces over (iii) Show that Tr/F, is surjective (iv) Let BE F....
8. Define (n) to be the number of positive integers less than n and n. That is, (n) = {x e Z; 1 < x< n and gcd(x, n) = 1}|. Notice that U (n) |= ¢(n). For example U( 10) = {1, 3,7, 9} and therefore (10)= 4. It is well known that (n) is multiplicative. That is, if m, n are (mn) (m)¢(n). In general, (p") p" -p Also it's well known that there are relatively prime, then...
Let Y_1~Gamma(α=3,β=3), Y_2~Gamma(α=5,β=1), and W=2Y_1+6Y_2.
a) (9 pts) Find the moment generating function ofW Justify all steps b) (3 pts) Based on your result in part (a), what is the distribution of W(name and parameters)? n 2N(O, I) 2. IfZ NO, 1), then Ux(1) 3. ItY Gmmaa,B) and W then Wx(n) - s, and i-1 7. y's~ Poisson(W (i-l, ,Rind) and U-ŽYi, then U-Poisson(XA) 8 If%-Gamma(a, β) (i-I, ,Rind) and U-ΣΥί , then U~Gamma( ,4 β).(Note: all same β) 9...
9. Let E be an extension field of a field F. (1) What does it mean for an element z EE being algebraic over F? (2) What does it mean for an element z E E being transcendental over F? (3) Can you find any element r e C such that r is transcendental over Q? (4) Can you prove that if a E E is algebraic over F then (F(a): F] is finite? (5) Can you prove that 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...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the field Q. Consider z E Q(w). Find a, b, c, d e Q us + w-2 such that : a + bu + cu2 + du 9. Let E be an extension field of a field F. (1) What does it mean for an element z E E being algebraic over F? (2) What does it mean for an element z EF being transcendental...
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).
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