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ID: past-exam-of-the-rings-and-modules-course-of-the-university-of-oxford/2017/1/b/iv/solution

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Past exam of the rings and modules course of the University of Oxford / 2017 / 1 / b / iv / Solution by

Codex  0 2026-10-31

R = F_{2} [t] / (t^{2} + t + 1) .

(1)

Write

α

for the class of

t

. Monic polynomial division over a ring gives a unique representative

a + b α

, with

a, b \in F_{2}

: existence follows by division, and uniqueness follows because a nonzero multiple of the degree-two modulus cannot have degree less than two. The four elements are therefore

0, 1, α, α + 1

The defining relation gives

α^{2} = α + 1

, and hence

α (α + 1) = 1

. Thus the nonzero elements have inverses

1^{- 1} = 1, α^{- 1} = α + 1, (α + 1)^{- 1} = α .

(2)

The quotient ring is commutative and every nonzero element is a unit, so it is a field. This verifies the field property directly, without quoting a theorem about irreducible-polynomial quotients.

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