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Let $\mathbb{R}[x]$ be the ring of polynomials with real coefficients in the determinate $x$. Each $\mathbb{R}$-algebra homomorphism from $\mathbb{R}[x]$ to $\mathbb{R}$ has the form $$ f(x) \mapsto f(\lambda) $$ for some $\lambda \in \mathbb{R}$.

Is there a (commutative unital) ring homomorphism $\varphi: \mathbb{R}[x]\to \mathbb{R}$ not of the above form?

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    use two things: what are the maximal ideals of $\mathbb R[x]$ and that the field $\mathbb R$ has no non-trivial automorphisms – 2017-02-11
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    Does $\mathbb{R}$ have a non-identity endomorphism? – 2017-02-12
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    No notrivial endomorphisms either (they have to be the identity on the rational numbers and by a simple argument they have to preserve the order) – 2017-02-12

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