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            <note type="commentary">New elementary criterion for the norm of the fundamental unit of a real quadratic field - 16 pages</note>
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                <term xml:lang="en">Real quadratic fields</term>
                <term xml:lang="en">Fundamental unit</term>
                <term xml:lang="en">Norm of units</term>
                <term xml:lang="en">Class field theory</term>
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              <p>We give an elementary criterion for the norm of the fundamental unit ε_K of K=Q(M^1/2), M square-free. More precisely, if ε_K = a+b M^1/2, a, b ∈ Z or 1/2Z, its norm S only depends on m := gcd[(a+1)/(\gcd(a+1,b)),M] and m' := gcd[(a-1)/(\gcd(a-1,b)),M] as follows when -1 is a global norm in K/Q: S = -1 if and only if m=m'=1 (resp. m=m'=2) for M odd (resp. even) (Theorems 1.1 or 2.4). Some statistics are suggested.</p>
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