only Part b nedded 3. Suppose p is prime and E is a field extension of Zp. Suppose there is a e E which is a zero ofxP – x + 1.(a) Prove that xP – x +1 = (x – a) · · · (x – a – p+ 1).(b) Prove that ma,z,(x) = xP – x+1. (Hint. Use part (a) and ma,Z,(x)|xP – x + 1.)(c) Deduce that æP – x +1 is irreducible in Zp[r].

Given: p is prime and E is field extension of Zp. α∈ E is a root of xp -x+1. (b): To prove: mα,zpx…

3. Suppose p is prime and E is a field extension of Zp. Suppose there is a € E which is a zero of rP – x+ 1. (a) Prove that xP – x + 1 = (x – a)· …· (x – a – p + 1). (b) Prove that ma,z,(x) = xº – x + 1. (Hint. Use part (a) and ma̟z,(x)|xP – x + 1.) (c) Deduce that xP – x +1 is irreducible in Z,[r].

3. Suppose p is prime and E is a field extension of Zp. Suppose there is a € E which is a zero of rP – x+ 1. (a) Prove that xP – x + 1 = (x – a)· …· (x – a – p + 1). (b) Prove that ma,z,(x) = xº – x + 1. (Hint. Use part (a) and ma̟z,(x)|xP – x + 1.) (c) Deduce that xP – x +1 is irreducible in Z,[r].

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 17E: Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive...

See similar textbooks

Similar questions

8. Prove that the characteristic of a field is either 0 or a prime.
Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].
Suppose S is a subset of an field F that contains at least two elements and satisfies both of the following conditions: xS and yS imply xyS, and xS and y0S imply xy1S. Prove that S is a field. This S is called a subfield of F. [Type here][Type here]
Use Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .
Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
Since this section presents a method for constructing a field of quotients for an arbitrary integral domain D, we might ask what happens if D is already a field. As an example, consider the situation when D=5. a. With D=5, write out all the elements of S, sort these elements according to the relation , and then list all the distinct elements of Q. b. Exhibit an isomorphism from D to Q.
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
Each of the polynomials in Exercises is irreducible over the given field . Find all zeros of in the field obtained by adjoining a zero of to . (In Exercises and , has three zeros in .)
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 . , ,
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.
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.
Consider the set ={[0],[2],[4],[6],[8]}10, with addition and multiplication as defined in 10. a. Is R an integral domain? If not, give a reason. b. Is R a field? If not, give a reason. [Type here][Type here]