4. Inside the ring R[×], consider the following ideal from Question 3 I = {h(x) = R[x] : h(0) = 0 and h(1) = 0} which consists of all polynomials h(x) which have a zero at x = 0 and at x = 1. Observe that x(x − 1) = x²xЄI. Using the results of Lecture 17, show that I = (x² - x) Hint: Show that x2 - xЄ Ihas smallest possible degree. Then appeal to a theorem Here is Question 3 3. In each case below, apply the extended Euclidean algorithm to a, bЄ Z. Determine d = gcd(a, b), and find integers x, y Z satisfying the Bézout identity: ax + by = d Finally, determine whether or not 5 is a unit in Z/aZ. If it is a unit, provide its multiplicative inverse by using the Bézout identity. (a) a = 27, b = 8 (b) a = 34, b = 10 (c) a = 102, b = 33 Here is the theorem Theorem. F[x] such that Let F be a field, and I ℃ F[x] an ideal. Then there exists a polynomial g(x) € I = (g(x)) = {multiples of g(x)} = {f(x) = F[x] : g(x)|f(x)} If I is non-trivial, then g(x) = I is a non-zero element of smallest possible degree.

4. Inside the ring R[×], consider the following ideal from Question 3 I = {h(x) = R[x] : h(0) = 0 and h(1) = 0} which consists of all polynomials h(x) which have a zero at x = 0 and at x = 1. Observe that x(x − 1) = x²xЄI. Using the results of Lecture 17, show that I = (x² - x) Hint: Show that x2 - xЄ Ihas smallest possible degree. Then appeal to a theorem Here is Question 3 3. In each case below, apply the extended Euclidean algorithm to a, bЄ Z. Determine d = gcd(a, b), and find integers x, y Z satisfying the Bézout identity: ax + by = d Finally, determine whether or not 5 is a unit in Z/aZ. If it is a unit, provide its multiplicative inverse by using the Bézout identity. (a) a = 27, b = 8 (b) a = 34, b = 10 (c) a = 102, b = 33 Here is the theorem Theorem. F[x] such that Let F be a field, and I ℃ F[x] an ideal. Then there exists a polynomial g(x) € I = (g(x)) = {multiples of g(x)} = {f(x) = F[x] : g(x)|f(x)} If I is non-trivial, then g(x) = I is a non-zero element of smallest possible degree.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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