Question

If T is an endomorphism of finite order, i.e, T^n=1 for some n>=1, give a necessary and sufficient condition on the field F for T to be diagonalizable

Answer #1

Prove that for any positive integer n, a field F can have at
most a finite number of elements of multiplicative order at most
n.

Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...

4) Let F be a finite field. Prove that there exists an integer n
≥ 1, such that n.1F = 0F . Show further that the smallest positive
integer with this property is a prime number.

Q)
Let L is the Laplace transform.
(i.e. L(f(t)) = F(s))
L(tf(t)) = (-1)^n * ((d^n)/(ds^n)) * F(s)
(a) F(s) = 1 / ((s-3)^2), what is f(t)?
(b) when F(s) = (-s^2+12s-9) / (s^2+9)^2, what is f(t)?

Master Theorem: Let T(n) = aT(n/b) + f(n) for some constants a ≥
1, b > 1.
(1). If f(n) = O(n logb a− ) for some constant > 0, then T(n)
= Θ(n logb a ).
(2). If f(n) = Θ(n logb a ), then T(n) = Θ(n logb a log n).
(3). If f(n) = Ω(n logb a+ ) for some constant > 0, and
af(n/b) ≤ cf(n) for some constant c < 1, for all large n,...

For an abelian group G, let tG = {x E G: x has finite order}
denote its torsion subgroup.
Show that t defines a functor Ab -> Ab if one defines t(f) =
f|tG (f restricted on tG) for every homomorphism f.
If f is injective, then t(f) is injective.
Give an example of a surjective homomorphism f for which t(f)
is not surjective.

Suppose n = 10. Is there some finite alphabet Q such that there
are vectors x, y, z, and w in Q^n (i.e., q-ary n-tuples, with q the
size of Q) such that (with d denoting Hamming distance) d(x,y) = 4,
d(x,z) = 8, d(x,w) = 5, d(y,z) = 3, d(y,w) = 4, and d(z,w) = 3 ?
(Hamming Distance)

Let G be a finite group and let H be a subgroup of order n.
Suppose that H is the only subgroup of order n. Show that H is
normal in G.
Hint: Consider the subgroup aHa-1 of G.
Please explain in detail!

True/False, explain:
1. If G is a finite group and G28, then there is a subgroup of G of
order 2401=74
2. If |G|=19, then G is isomorphic to Z19.
3. If F subset of K is a degree 5 field extension, any element b in
K is the root of some polynomial p(x) in F[x]
4. If F subset of K is a degree 5 field extension, viewing K as
a vector space over F, Aut(K, F) consists of...

Consider the sequence(an)n≥1that starts1,3,5,7,9,...(i.e, the
odd numbers in order).
(a) Give a recursive definition and closed formula for the
sequence.
(b) Write out the sequence(bn)n≥2 of partial sums of (an). Write
down the recursive definition for (bn) and guess at the closed
formula.
(b) How did you get the partial sums?

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