We’ll say a polynomial f(x) ∈ R[x] is prime if the ideal (f(x)) ⊂ R[x] is...

We’ll say a polynomial f(x) ∈ R[x] is prime if the ideal (f(x)) ⊂ R[x] is prime. If F is a field with finitely many elements (e.g., Z/pZ), prove that f(x) ∈F [x] is prime if and only if it’s irreducible.

Expert Solution

Colby Messinger answered 2 years ago

f: R[x] to R is the map defined as f(p(x))=p(2) for any polynomial p(x) in R[x]....

f: R[x] to R is the map defined as f(p(x))=p(2) for any polynomial p(x) in R[x]. show that f is 1) a homomorphism 2) Ker(f)=(x-2)R[x] 3) prove that R[x]/Ker(f) is an isomorphism with R. (R in this case is the Reals so R[x]=a0+a1x+a1x^2...anx^n)

let R = Z x Z. P be the prime ideal {0} x Z and S...

let R = Z x Z. P be the prime ideal {0} x Z and S = R - P. Prove that S^-1R is isomorphic to Q.

Prove the following: Let f(x) be a polynomial in R[x] of positive degree n. 1. The...

Prove the following: Let f(x) be a polynomial in R[x] of positive degree n. 1. The polynomial f(x) factors in R[x] as the product of polynomials of degree 1 or 2. 2. The polynomial f(x) has n roots in C (counting multiplicity). In particular, there are non-negative integers r and s satisfying r+2s = n such that f(x) has r real roots and s pairs of non-real conjugate complex numbers as roots. 3. The polynomial f(x) factors in C[x] as...

Let R be a commutative ring with unity. If I is a prime ideal of R,...

Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x].

Let R be a commutative domain, and let I be a prime ideal of R. (i)...

Let R be a commutative domain, and let I be a prime ideal of R. (i) Show that S defined as R \ I (the complement of I in R) is multiplicatively closed. (ii) By (i), we can construct the ring R1 = S-1R, as in the course. Let D = R / I. Show that the ideal of R1 generated by I, that is, IR1, is maximal, and R1 / I1R is isomorphic to the field of fractions of...

Use induction to prove Let f(x) be a polynomial of degree n in Pn(R). Prove that...

Use induction to prove Let f(x) be a polynomial of degree n in Pn(R). Prove that for any g(x)∈Pn(R) there exist scalars c0, c1, ...., cn such that g(x)=c0f(x)+c1f′(x)+c2f′′(x)+⋯+cnf(n)(x), where f(n)(x)denotes the nth derivative of f(x).

Given n ∈N and p prime number and consider the polynomial f (x) = xn (xn-2)+1-p...

Given n ∈N and p prime number and consider the polynomial f (x) = xn (xn-2)+1-p 1)Prove that f (x) is irreducible in Q [x] 2) If n = 1 and p = 3, find Q [x] / f (x)) 3) Show that indeed Q [x] / (f (x)) is a field in the previous paragraph PLEASE answer all subsections

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We’ll say a polynomial f(x) ∈ R[x] is prime if the ideal (f(x)) ⊂ R[x] is...

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