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.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.4: Zeros Of A Polynomial
Problem 19E
Related questions
Question
Could you solve the question 4, please?
I have attached question 3 and the theorem in the screenshot
Thank you.
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5614-e5fa-4399-aabc-c345eeef0588%2F78ee39ca-d84e-42f0-9a8c-09098148a9fd%2Fu1xpxt_processed.png&w=3840&q=75)
Transcribed Image Text: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.
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