What is the example of a Noetherian local ring of dimension one which is not a discrete valuation ring.
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Take the local ring of any curve singularity, for instance localize $k[x,y]/(x^2y^3)$ at $(x,y)$.
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Take the ring of polynomials over a noetherian local ring with Krull dimension 0, say $\mathbf Z/p^2\mathbf Z$, localised at the maximal ideal $\langle (p\mathbf Z/p^2\mathbf Z, X\rangle$. Or the ring of formal power series over the same ring.

I suppose that "integral domain" is somehow implicit. – user26857 May 22 '17 at 13:31