8. Consider the Galois field GF(24), with standard addition, and multiplication defined modulothe (primitive, thus irreducible) polynomial P (x) = x4 + x + 1. Over this field, compute theresult of multiplying the elements 1101 and 0110.

Galois field denoted by GF(q=pn)is a field with characteristic p and a number q of elements. Here…

Answered: Consider the Galois field GF(24), with…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.6: Algebraic Extensions Of A Field

Problem 2TFE

See similar textbooks

Similar questions

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.
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 .
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.
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 . , ,
Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.
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 .)
True or False Label each of the following statements as either true or false. Every polynomial equation of degree over a field can be solved over an extension field 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.
If is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .
Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero in
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]

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

Consider the Galois field GF(24), with standard addition, and multiplication defined modulo the (primitive, thus irreducible) polynomial P (x) = x4 + x + 1. Over this field, compute the result of multiplying the elements 1101 and 0110. %3D