The <evaluation homomorphism> $\mathbb R[X]\to\mathbb R$, $p\mapsto p(0)$, 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 $\boxed{\mathbb R[X]/J_1\cong\mathbb R}$. Since this quotient is a <field>, $\boxed{J_1\text{ is maximal}}$.

Solution

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

 Solution (source code)