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ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2006/3/i/solution

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Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2006 / 3 / i / Solution by

Codex  0 2026-10-31

The evaluation homomorphism

R [X] \to R

,

p \mapsto p (0)

, is surjective and has kernel of a ring homomorphism

J_{1} = (X)

. Hence

J_{1}

is an ideal, and the first isomorphism theorem for rings gives

R [X] / J_{1} ≅ R

. Since this quotient is a field,

J_{1} is maximal

.

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