find out the question is true or false and give explanations (4).(5).mial ring C[x].(6).The ring F[x]/(x²) has a unique maximal ideal.The set {fe C[x] | ƒ(2) = 0, ƒ (0) = 0} is a principal ideal of the polyno-The field Q[√5] is the field of quotients of Z[√-5]

4)False The statement "The ring F[x]/(x^2) has a unique maximal ideal" is false. Explanation: A…

Answered: (4). (5). mial ring C[x]. (6). The ring…

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 7TFE

See similar textbooks

Similar questions

8. Prove that the characteristic of a field is either 0 or a prime.
Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.
18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
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.
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 that if R is a field, then R has no nontrivial ideals.
Prove Theorem If and are relatively prime polynomials over the field and if in , then 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 .
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 .
15. Let and be elements of a ring. Prove that the equation has a unique solution.
a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].
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 .)

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

(4). (5). mial ring C[x]. (6). The ring F[x]/(x²) has a unique maximal ideal. The set {fe C[x] | ƒ(2) = 0, ƒ (0) = 0} is a principal ideal of the polyno- The field Q[√5] is the field of quotients of Z[√-5]