Let F be a commutative ring with identity and let f (x) E F[x] be_awwwpolynomial of degree 4. Which of the following statements is true:f(x) has exactly 4 roots countingmultiplicity.f(x) has exactly 4 distinct roots Xf(x) has at most 4 roots.None

Answered: Image /qna-images/answer/42bf912f-b52b-40b3-91a3-c0ea9891f87a.jpg

Answered: Let F be a commutative ring with…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 18E: 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is...

See similar textbooks

Similar questions

A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.
46. Let be a set of elements containing the unity, that satisfy all of the conditions in Definition a, except condition: Addition is commutative. Prove that condition must also hold. Definition a Definition of a Ring Suppose is a set in which a relation of equality, denoted by , and operations of addition and multiplication, denoted by and , respectively, are defined. Then is a ring (with respect to these operations) if the following conditions are satisfied: 1. is closed under addition: and imply . 2. Addition in is associative: for all in. 3. contains an additive identity: for all . 4. contains an additive inverse: For in, there exists in such that . 5. Addition in is commutative: for all in . 6. is closed under multiplication: and imply . 7. Multiplication in is associative: for all in. 8. Two distributive laws hold in: and for all in . The notation will be used interchageably with to indicate multiplication.
Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .
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 ].
[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]
18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]
Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.
Find all monic irreducible polynomials of degree 2 over Z3.

SEE MORE QUESTIONS

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

Let F be a commutative ring with identity and let f (x) E F[x] be a polynomial of degree 4. Which of the following statements is true: f(x) has exactly 4 roots counting multiplicity. f(x) has exactly 4 distinct roots X f(x) has at most 4 roots. None < >