3.Determine whether the following mappings are ring homomorphisms:f: Q- Q defined by f(x)= |x| for all x e Q.a)b)c)g: CM₂ (R) defined by g (a + bi) =ab-b ah: Z[√2] → Z[√2] defined by h( a + b √2) =) a - b √2

Answered: 3. Determine whether the following…

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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Please help with both parts b and c: (will leave a like)
(a) Consider the function f : R × R → C defined by f ((a, b)) = a + bi. Prove or disprove whether f is an isomorphism.(b) Consider the function g : R∗ × R∗ → C∗ defined by f ((a, b, )) = a + bi. Prove or disprove whether g is an isomorphism
Let G = Z16X Z4. (i) Construct a surjective homomorphism : G → Z8. (ii) What is ker()? (iii) Show that there is no surjective homomorphism y: G → Z8 × Z8.
Let D: Q[x] → Q[x] akx²+...+a₁x + að ↔ kakxk−1 +...+a₁ be the derivative map on rational polynomials. Is D a ring homomorphism? Is it an isomorphism?
(a) Let u and v be two vectors in R³ such that ||u|| = 4 and ||v|| possible values of u · v and ||u – v||. (b) Let v = (2, -3,4), u = = (4, 2, −3) € R³, find all real value(s) of c such that (v-cu) will be orthogonal to u. = 7, find the largest and smallest
Define a ring homomorphism 4: R[x] → RX R, (ƒ(x)) = (fƒ(0), ƒ(1)) In other words, y maps f(x) to its evaluations at x = and at x = = 1. (a) Describe the kernel of y, in terms of polynomials having certain roots/zeroes. (b) Find polynomials f(x) and g(x) in R[x] such that f(0) = 1, f(1) = 0, and g(0) = 0, g(1) = 1 (Note: There are many different options!) Let a, b € R, and define h(x) = af (x) + bg(x). Show that y(h(x)) = (a,b). (c) Using part (b), explain why is surjective. (d) What does the 1st Isomorphism Theorem say, when applied to ?
(19) Let om : Z → Zm be the ring homomorphism defined by om (a) remainder of division of a by m. (a) Show that m : Z[x] → Zm[x] defined by om (anx + + a₁x + ao) = om(an)x² + is a ring homomorphism onto Zm[x]. + om (a₁)x+om (ao) = (b) Show that if f(x) = Z[x] and ¯m (f(x)) = Zm[x] both have degree n and σm(f(x)) does not factor in Zm[x] into two polynomials of degree less than n, then f(x) is irreducible Q[x]. (c) Use the previous part to show that x³ + 17x + 36 is irreducible in Q[x].
Abstract algebra 2. Graduate level. Note: let R be any ring and M and N R − modules

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3. Determine whether the following mappings are ring homomorphisms: f: Q- Q defined by f(x)= |x| for all x e Q. a) b) c) g: CM₂ (R) defined by g (a + bi) = a b -b a h: Z[√2] → Z[√2] defined by h( a + b √2) =) a - b √2