7. Answer part (a) and part (b):(a) In the ring Z[x] of polynomials with integer coefficients, determineif the following ideals are Maximal, Prime but not Maximal, or notPrime. You do not have to provide reasonsi. (the ideal generated by x)ii. <²> (the ideal generated by x²)iii. <2, x> (the ideal generated by 2 and x)iv. < x,x+1> (the ideal generated by x and x + 1)(b) Are the following rings PIDs, a UFD but not a PID, or not a UFD?You do not have to provide reasonsi. Q[x, y] (the ring in two indeterminates over the field of rationalnumbers)ii. Zp [x] (the polynomial ring in one indeterminate over the field ofintegers modulo a prime p)iii. Z [√-5]

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Answered: if the following ideals are Maximal,…

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 12E: a. Find a nonconstant polynomial in Z4[ x ], if one exists, that is a unit. b. Find a nonconstant...

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18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
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 ].
Exercises If and are two ideals of the ring , prove that is an ideal of .
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 .
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. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
Prove that [ x ]={ a0+a1x+...+anxna0=2kfork }, the set of all polynomials in [ x ] with even constant term, is an ideal of [ x ]. Show that [ x ] is not a principal ideal; that is, show that there is no f(x)[ x ] such that [ x ]=(f(x))={ f(x)g(x)g(x)[ x ] }. Show that [ x ] is an ideal generated by two elements in [ x ] that is, [ x ]=(x,2)={ xf(x)+2g(x)f(x),g(x)[ x ] }.
33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .)
a. Find a nonconstant polynomial in Z4[ x ], if one exists, that is a unit. b. Find a nonconstant polynomial in Z3[ x ], if one exists, that is a unit. c. Prove or disprove that there exist nonconstant polynomials in Zp[ x ] that are units if p is prime.
Show that the ideal is a maximal ideal of .
Let Q denote the field of rational numbers, R the field of real numbers, and C the field of complex. Determine whether each of the following polynomials is irreducible over each of the indicated fields, and state all the zeroes in each of the fields. a. x22 over Q, R, and C b. x2+1 over Q, R, and C c. x2+x2 over Q, R, and C d. x2+2x+2 over Q, R, and C e. x2+x+2 over Z3, Z5, and Z7 f. x2+2x+2 over Z3, Z5, and Z7 g. x3x2+2x+2 over Z3, Z5, and Z7 h. x4+2x2+1 over Z3, Z5, and Z7
Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .

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