Question

Prove this statement. Z[i] is the Gaussian integers:

Let α ∈ Z[i]. Then α is a unit if and only if N(α) = 1.

Answer #1

Due October 25. Let Z[i] denote the Gaussian integers, with norm
N(a + bi) = a 2 + b 2 . Recall that ±1, ±i are the only units i
Z[i]. (i) Use the norm N to show that 1 + i is irreducible in Z[i].
(ii) Write 2 as a product of distinct irreducible elements in
Z[i].

Let Z be the integers.
(a) Let C1 = {(a, a) | a ∈ Z}. Prove that
C1 is a subgroup of Z × Z.
(b) Let n ≥ 2 be an integer, and let Cn = {(a, b) | a
≡ b( mod n)}. Prove that Cn is a subgroup of Z × Z.
(c) Prove that every proper subgroup of Z × Z that contains
C1 has the form Cn for some positive integer
n.

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

Prove: Let n ∈ N, a ∈ Z, and gcd(a,n) = 1. For i,j ∈ N,
aj ≡ ai (mod n) if and only if j ≡ i (mod
ordn(a)). Where ordn(a) represents the order
of a modulo n. Be sure to prove both the forward and backward
direction.

Prove: Let a and b be integers. Prove that integers a and b are
both even or odd if and only if 2/(a-b)

Suppose n and m are integers. Let H = {sm+tn|s ∈ Z and t ∈
Z}.
Prove that H is a cyclic subgroup of Z.
......................
Please help with clear steps that H is a cyclic subgroup of
Z

Let two independent random vectors x and z have Gaussian
distributions: p(x) = N(x|µx,Σx), and p(z) = N(z|µz,Σz). Now
consider y = x + z. Use the results for Gaussian linear system to
ﬁnd the distribution p(y) for y. Hint. Consider p(x) and p(y|x).
Please prove for it rather than directly giving the result.

Let Z be the set of integers. Define ~ to be a relation on Z by
x~y if and only if |xy|=1. Show that ~ is symmetric and transitive,
but is neither reflexvie nor antisymmetric.

Kernels on real vectors
Let x,z ∈ Rn, show the following is valid kernel:
Gaussian or RBF: k(x, z) = exp(-α ||x - z||2), for α
> 0.

. Prove that, for all integers n ≥ 1, Pn i=1 i(i!) = (n + 1)! −
1

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