Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant$$\Delta =7^2,9^2,{13}^2,{19}^2,{31}^2,{37}^2,{43}^2,{61}^2,{67}^2,{103}^2,{109}^2,{127}^2,{157}^2.$$A large part of the proof is in establishing the following more general result: Let $K$ be a Galois number field of odd prime degree $\ell$ and conductor $f$. Assume the GRH for ${\zeta }_K\left(s\right)$. If$$38{(\ell -1)}^2{(logf)}^{6}loglogf\<f,$$then $K$ is not norm-Euclidean.

Some quadratic forms related to "greatest common divisor matrices" are represented in terms of L²-norms of rather simple functions. Our formula is especially useful when the size of the matrix grows, and we will study the asymptotic behaviour of the smallest and largest eigenvalues. Indeed, a sharp bound in terms of the zeta function is obtained. Our leading example is a hybrid between Hilbert's matrix and Smith's matrix.