**Consider the Substitution Homomorphism** Given the homomorphism \( M_{t_2} : \mathbb{Z}[x] \to \mathbb{Z}(t_2) \) defined by \( M_{t_2}(p(x)) = p(t_2) \).(You can assume it's a homomorphism.)(a) Show \( M_{t_2} \) is onto.(b) Determine \( \text{Ker } M_{t_2} \) and prove your claim.(c) Is \( \text{Ker } M_{t_2} \) prime, maximal, or neither? Why?

Given : μ2 : ℤx → ℤ2 defined by μ2p(x) = p2 is a homomorphism. a) Show that μ2 is onto. b)…

Answered: (9) Consider the substitution…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1 2 3 4 5 6 3 6 4 51 2 1 2 3 4 56 2 4 6 5 3 1)! Consider permutations o = and r = in S6. Find: (a) composition too; (b) a-. (b) Determine the domain for each of the following functions. 1 1.) f(2)- 1 4.) m (x) = V – 6x 7.) f(x) + Vr? - 4x + 3 I+5 12 – 25 I +7 1² +1 1. 2.) g(x) = VI + 5 5.) f (1) = 8.) g(z) = Vr? + 4- %3D %3D %3D Va +2–3 3.) p(z) 6.) t(r). 9.) f(x) = Vr -1+ v0 %3D 2 – 6x
Number 5
I need the answer as soon as possible
8 6. Let A=|3 5 7 | 4 9 (i) Evaluate det(A) by cofactor expansion. 2 (ii) If A is invertible, find its inverse by using the adjoint of A.
Why i(3^0) = i(3^8) = 0.please with explanation. I think 3^0 is always 0 . (3 is the notation , not number)
Please answer it quickly.
Q1
3. Determine the Jordan Canonical Form of A if the characteristic polynomial of A is given by (1-A)²(2-x)³ with dim(Ker(A-I)) = 1 and dim(Ker(A- 21)) = 2.
Answer letter A only.
13) The integer n = a) If n = 3 (mod 9), then what is x ? b) If n = 3 (mod 11), then what is x ? Hint for #13: 12700140x3243243 is missing the digit x. Consider the number 1234. By writing each digit in scientific notation, we have 1234 1 x 10³ +2× 10² +3 × 10¹ + 4 × 10⁰ Then notice that 10 = 1 (mod 9). When we reduce mod 9, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x 1³ + 2 x 1² + 3 × 1¹ + 4 × 1⁰ = 1+ 2+ 3+ 4 = 10 = 1 (mod 9). Since 10 = -1 (mod 11), when we reduce mod 11, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x (-1)³ +2× (-1)² +3 × (-1)¹ +4× (-1)⁰ = -1 + 2-3 + 4 = 2 (mod 11).
(9) Let f(r) be an Irreducible cubic over Q with cyclic Galols group. Show that all roots of f(z) are real.
4. Let w = e 11, and let E = Q(w). (a) Verify that [E:Q] 10. (b) Determine |Gal(E/Q)|, the number of automorphisms of E that fix Q. (c) Find the order of y(w) w? in Gal(E/Q). (d) How many subfields K does E have (such that QC K C E)? Hint 1 The corollary to Eisenstein's theorem might help. Hint 2: Example 6, page 535 may help.

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(9) Consider the substitution homomorphism given by My (Peas) = p (6₂) (You can assume it's a homonoephium) M: 2[^) -> 2 [r] (a) Show M₂ is onto (6) Determine Kerfly Prove your claim. (c) Is Kerl, Prime, Meximel or neither? Why Kerlly and