only if {0} is a prime ideal? (m) If I = (x³-2x) is an ideal of Z[x], then what is a polynomial p(x) = Z[x] that has as low a degree as possible and such that x6 + x5 + I = p(x) + I? (n) Let F3 (Z/3Z, +,), R = F3[x], and I = (x3 + x + 2). Is R/I a field with 27 elements? = (o) Let F3 = (Z/3Z, +,), R = F3[x], and I = (x3 + 2x + 1). Is R/I a field with 27 elements?

Q: Let F = Q, s(x) = x² + 1, and E = Q[r]/(x² + 1)Q[x]. i) Factor the polynomial r? +1€ Q[r] into…

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Q: Consider a polynomial pn(x) = ao + a₁x +...+ anx" = [ao a₁ z. Synthetic division gives us an] (x)…

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A: Finding the values of b and no

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Q: If R=Z[x] and f(x) = x2 + 1, remain true, but g(x) =x. How do you prove that in R/I, [g(x)] x [g(x)]…

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Q: 1. Consider y=f(x)%3Dsin(3x)+x² over [0, 37/2]. (By hand but use a calculator) a) Use the three…

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Q: Let p be a prime and let q = p". Let f(x) = x³+ax+b € Fq[x] be

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Q: Let f(x) = x³ + x² - 4x − 4, g(x) = x¹ + 3x³ + 5x² +9x + 6 and consider the ideal I = (f(x), g(x))…

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Q: Check if f (x) = 4x³ + 3x¹ − 12x² + 3 is irreducible in Q[r].

A: Note: As company policy, we solve only the first question. Please resubmit the rest of the questions…

Q: an

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Q: Which of the following subsets in the set of polynomials R[x] is not closed for the operation of…

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Q: d) Prove: If Z = Zo is a zero of order n of flz), then it is zero of order 2n of (f(z))²

A: