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Compute in a polynomial ring


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.1

Let $p(x) = 2x^3 - 3x^2 + 4x - 5$ and let $q(x) = 7x^3 + 33x - 4$. In each of parts (a), (b), and (c), compute $p+q$ and $pq$ under the assumption that the coefficients of the two given polynomials are taken from the specified ring (where the integer coefficients are taken mod $n$ in parts (b) and (c)): (a) $R = \mathbb{Z}$, (b) $R = \mathbb{Z}/(2)$, (c) $R = \mathbb{Z}/(3)$.


Solution:

(a) Evidently, $$p(x)+q(x) = 9x^3 - 3x^2 + 37x - 9$$ and $$p(x)q(x) = 14x^6 - 21x^5 + 94x^4 - 142 x^3 + 144 x^2 - 181 x + 20.$$(b)In $\mathbb{Z}/(2)$, $p(x) = x^2 + 1$ and $q(x) = x^3 + x$. Evidently, $$p(x) + q(x) = x^3 + x^2 + x + 1$$ while $$p(x)q(x) = x^5 + x.$$(c) In $\mathbb{Z}/(3)$, $p(x) = 2x^3 + x + 1$ and $q(x) = x^3 + 2$. Evidently, $p(x) + q(x) = x$ while $$p(x)q(x) = 2x^6 + x^4 + + 2x^3 2x + 2.$$


Linearity

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