5.16. Let F be a field, and suppose that the polynomial X² + X + 1 is irreducible in F[X]. Let K = F[X]/(X² + X + 1)F[X] be the quotient ring, so we know from Theorem 5.26 that K is a field. We will put bars over polynomials to indicate that they represent elements of K. In other words, if we let I be the ideal I = (X² + X + 1)F[X], then X + 2 is shorthand for the coset (X + 2) + I. (a) Find a polynomial p(X) = F[X] of degree at most 1 satisfying p(X) = (X + 3) · (2X + 1). (b) Find a polynomial q(X) € F[X] of degree at most 1 satisfying q(X) · (X + 1) = 1. In other words, find a multiplicative inverse for X + 1 in the field K. (c) Find a polynomial r(X) € F[X] of degree at most 1 satisfying
5.16. Let F be a field, and suppose that the polynomial X² + X + 1 is irreducible in F[X]. Let K = F[X]/(X² + X + 1)F[X] be the quotient ring, so we know from Theorem 5.26 that K is a field. We will put bars over polynomials to indicate that they represent elements of K. In other words, if we let I be the ideal I = (X² + X + 1)F[X], then X + 2 is shorthand for the coset (X + 2) + I. (a) Find a polynomial p(X) = F[X] of degree at most 1 satisfying p(X) = (X + 3) · (2X + 1). (b) Find a polynomial q(X) € F[X] of degree at most 1 satisfying q(X) · (X + 1) = 1. In other words, find a multiplicative inverse for X + 1 in the field K. (c) Find a polynomial r(X) € F[X] of degree at most 1 satisfying
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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![5.16. Let F be a field, and suppose that the polynomial X² + X + 1 is irreducible in F[X]. Let
K = F[X]/(X² + X + 1)F[X]
be the quotient ring, so we know from Theorem 5.26 that K is a field. We will put bars over
polynomials to indicate that they represent elements of K. In other words, if we let I be the ideal
I = (X² + X + 1)F[X], then X + 2 is shorthand for the coset (X + 2) + I.
(a) Find a polynomial p(X) € F[X] of degree at most 1 satisfying
p(X) = (X+3) · (2X + 1).
(b) Find a polynomial q(X) € F[X] of degree at most 1 satisfying
q(X) · (X + 1) = I.
In other words, find a multiplicative inverse for X + 1 in the field K.
(c) Find a polynomial r(X) € F[X] of degree at most 1 satisfying
r(x)² =
= −3.
In other words, find a square root of −3 in the field K.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F081e66de-98dd-47c4-ad89-12bf6ce96be8%2Ffedc7b16-d6ae-47a6-ade6-07b5163066a1%2Fwmbtwyc_processed.png&w=3840&q=75)
Transcribed Image Text:5.16. Let F be a field, and suppose that the polynomial X² + X + 1 is irreducible in F[X]. Let
K = F[X]/(X² + X + 1)F[X]
be the quotient ring, so we know from Theorem 5.26 that K is a field. We will put bars over
polynomials to indicate that they represent elements of K. In other words, if we let I be the ideal
I = (X² + X + 1)F[X], then X + 2 is shorthand for the coset (X + 2) + I.
(a) Find a polynomial p(X) € F[X] of degree at most 1 satisfying
p(X) = (X+3) · (2X + 1).
(b) Find a polynomial q(X) € F[X] of degree at most 1 satisfying
q(X) · (X + 1) = I.
In other words, find a multiplicative inverse for X + 1 in the field K.
(c) Find a polynomial r(X) € F[X] of degree at most 1 satisfying
r(x)² =
= −3.
In other words, find a square root of −3 in the field K.
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