Math 149 HP M2/Two polynomial congruence anchor problems
9. A. Of the following two sets, one is a field and one is a ring. Explain in detail why or
why not for both of the sets using the mathematical justifications you have learned in
class:
i)
ℚ[𝑥]
⟨𝑥 4 −2𝑥 3 +2⟩
ii) ⟨𝑥 4
ℚ[𝑥]
+2𝑥 3 −2𝑥 2 +2𝑥−3⟩
B. For the set above that is a field, find [𝑥 + 1]−1 𝑝(𝑥) and demonstrate your answer is
indeed the multiplicative inverse modp(x).
Math 149 HP M2/Two polynomial congruence anchor problems
10. A. In ℤ5 [𝑥], find the degree-two polynomial of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 𝑦, running
through the following points: (1,3), (2, 4), (3, 4) using matrix reduction mod5. Discuss
why this polynomial is irreducible in ℤ5 [𝑥].
ℤ5 [𝑥]
, where p(x) is the polynomial you found above, find the
⟨𝑝(𝑥)⟩
multiplicative inverse for the element [𝑥 + 1] and demonstrate your solution is indeed the
inverse modp(x). (Remember to do your base-x divisions mod5 for the Euclidean algorithm).
B. In the field