Question

If T is an endomorphism of finite order, i.e, T^n=1 for some n>=1, give a necessary...

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

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

If is an endomorphism of finite order, then is diagonalisable if and only if is an algebraically closed field where the characteristic of the field does not divide the order of the endomorphism. This follows from the fact that is diagonalisable if and only if it's minimal polynomial in splits in and has distinct roots (the first condition gives algebraic closure, the second gives the condition for characteristic not dividing the order of endomorphism).

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Prove that for any positive integer n, a field F can have at most a finite...
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...
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,...
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))...
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 >...
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...
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,...
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...
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...
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...
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?
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT