For a field F find an irreducible polynomial f(x) which generates the same ideal in F[x] as the two polynomials x2 + x, x +3x+2 That means that the set of linear combinations F[x](x² + x) + F[ x](x² + 3x +2)= F[x ]f(x). This polynomial will be a linear combination of the two given polynomials times other polynomials,

For a field F find an irreducible polynomial f(x) which generates the same ideal in F[x] as the two polynomials x2 + x, x +3x+2 That means that the set of linear combinations F[x](x² + x) + F[ x](x² + 3x +2)= F[x ]f(x). This polynomial will be a linear combination of the two given polynomials times other polynomials,

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...

See similar textbooks

Similar questions

Prove Theorem If and are relatively prime polynomials over the field and if in , then in .
Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
Corollary requires that be a field. Show that each of the following polynomials of positive degree has more than zeros over where is not a field. over over
Let F be a field and f(x)=a0+a1x+...+anxnF[x]. Prove that x1 is a factor of f(x) if and only if a0+a1+...+an=0. Prove that x+1 is a factor of f(x) if and only if a0+a1+...+(1)nan=0.
True or False Label each of the following statements as either true or false. 8. Any polynomial of positive degree that is reducible over a field has at least one zero in .
True or False Label each of the following statements as either true or false. 4. Any polynomial of positive degree over the field has exactly distinct zeros in .
Let be a field. Prove that if is a zero of then is a zero of
Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.
In Exercises , a field , a polynomial over , and an element of the field obtained by adjoining a zero of to are given. In each case: Verify that is irreducible over . Write out a formula for the product of two arbitrary elements and of . Find the multiplicative inverse of the given element of . , ,
1. Find a monic polynomial of least degree over that has the given numbers as zeros, and a monic polynomial of least degree with real coefficients that has the given numbers as zeros. a. b. c. d. e. f. g. and h. and
Prove the Unique Factorization Theorem in (Theorem). Theorem Unique Factorisation Theorem Every polynomial of positive degree over the field can be expressed as a product of its leading coefficient and a finite number of monic irreducible polynomials over . This factorization is unique except for the order of the factors.
Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.