3) Show that the irreducible polynomials in R[T] are the following: (a) Linear polynomials, i.e. polynomials of degree 1; and (b) Quadratic polynomials, i.e. polynomials of degree 2, which do not have a real root. If g(T) is such a polynomial then g(T) = a · (T – z) · (T – z) = a · (T² + (z + z) · T + z• z), where z is a complex root of g(T), and a e R \ {0}.
3) Show that the irreducible polynomials in R[T] are the following: (a) Linear polynomials, i.e. polynomials of degree 1; and (b) Quadratic polynomials, i.e. polynomials of degree 2, which do not have a real root. If g(T) is such a polynomial then g(T) = a · (T – z) · (T – z) = a · (T² + (z + z) · T + z• z), where z is a complex root of g(T), and a e R \ {0}.
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 1E: 1. Find a monic polynomial of least degree over that has the given numbers as zeros, and a monic...
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![3) Show that the irreducible polynomials in R[T] are the following:
(a) Linear polynomials, i.e. polynomials of degree 1; and
(b) Quadratic polynomials, i.e. polynomials of degree 2, which do not have
a real root. If g(T) is such a polynomial then
g(T) = a · (T – 2) · (T – z) = a · (T² + (z + z) · T + z · z),
where z is a complex root of g(T), and a E R \ {0}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1bbb03a-1330-4abd-86a4-84230eb34f64%2Fe316fb20-8d9f-4692-a6ec-070c7ea2fa50%2Fnj0koq2_processed.png&w=3840&q=75)
Transcribed Image Text:3) Show that the irreducible polynomials in R[T] are the following:
(a) Linear polynomials, i.e. polynomials of degree 1; and
(b) Quadratic polynomials, i.e. polynomials of degree 2, which do not have
a real root. If g(T) is such a polynomial then
g(T) = a · (T – 2) · (T – z) = a · (T² + (z + z) · T + z · z),
where z is a complex root of g(T), and a E R \ {0}.
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