Let Z[X] be the ring of polynomials with integer coefficients and let å be the collection of all polynomials whose constant term is even. a. a is an ideal. b. a is a maximal ideal.

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A: As per the question we are given the polynomial ring ℤ[x] and its subset I defined as : I = {f(x) ∈…

Q: Prove that the degree of any irreducible factor of x 8 - x over Z2 is1 or 3.

A: We have given that, fx=x8-x We need to show that degree of any irreducible factor of fx over ℤ2 is 1…

Q: Let a, b e Z and let d:= (a, b). Prove that { ia + jb : i,j € Z}={kd : k E Z}. In other words, prove…

A: Let a,b∈ℤ and d=g.c.d.(a,b). The g.c.d.(a,b) divides a and b both. Therefore, ∃ m1,m2 such that…

Q: If a polynomial is reducible over Q then it is reducible over Z. O True O False

A: To show whether the given sentence is true or false.

Q: Give an example of a polynomial ring Rx and a polynomial of degree n with more than n zeros over R.

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Q: Prove that the degree of any irreducible factor of x8 - x over Z2 is 1 or 3.

A: Given the polynomial, fx=x8-x. To prove that the degree of any irreducible factor of x8-x over ℤ2 is…

Q: if the following ideals are Maximal, Prime but Prime. You do not have to provide reasons i. (the…

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Q: Let I = {(a, 0) | a e Z}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z.

A: Ideal of a ring

Q: Let R be a principal ideal domain. For any two distinct nonzero elements ƒ and g with no common…

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Q: Let I C R[X] be the subset of all polynomials f(X) E R[X] satisfying the condition f(2) = f(3).…

A: This question is related to ring theory . Solution is given below

Q: Let R[x] be the set of all expressions a = ao+ajx+a2x² + ……=£a;x ....- i=0 where a¡ E IR for all…

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Q: Find the minimal polynomial and the conjugates over Q of the algebraic V-3 number 5

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Q: Let α be a zero of an irreducible quadratic polynomial over Z5. Provethat there are elements a and b…

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