Dirichlet's unit theorem

Let K have signature $(r_1,r_2)$, with $[K:\mathbb Q]=r_1+2r_2$. Dirichlet's unit theorem gives an abstract isomorphism $\mathcal O_K^\times\cong\mu(K)\times\mathbb Z^{r_1+r_2-1}$, where μ(K) is the finite cyclic group of roots of unity in K. The theorem's rank counts multiplicatively independent units of infinite order. It does not say that the group has only that many elements or that an arbitrary list of units of the right length already generates the full group.

A real quadratic field has two real embeddings and no complex pair, hence free rank one. An imaginary quadratic field has one complex pair and no real embedding, hence rank zero and only finitely many units. Both have degree two, so degree alone is not enough to determine the unit rank. Roots of unity also depend on the actual field: the Gaussian integers contain four, while many other imaginary quadratic fields contain only plus and minus one. Those finite factors are independent of the free-rank count.

Choose all real embeddings σᵢ and one embedding τⱼ from each complex pair. Use log absolute values for the real entries and twice log absolute values for the complex entries. For a unit, their sum is the logarithm of the absolute rational norm, hence zero. Thus the logarithmic image lies in a real hyperplane of dimension $r_1+r_2-1$. Multiplication becomes vector addition. Omitting the factor two at a complex place changes the product formula and the hyperplane equation.

The kernel on integer-ring units is the finite root-of-unity group, and Dirichlet's theorem says that the image is a full lattice in that hyperplane. Discreteness and full rank are substantial conclusions, not consequences of the sum-zero equation alone. The covolume, with an appropriate convention, leads to the regulator. Choosing fundamental units amounts to choosing a lattice basis, so different bases can represent the same group. A finite-index sublattice of correct rank still misses units and gives an incorrect regulator for the full unit group.

In a real quadratic field, a candidate with norm plus or minus one is a unit, and its powers supply infinitely many units if its chosen real value exceeds one. To call it a fundamental unit requires showing that no smaller positive unit greater than one exists, or otherwise proving the generated subgroup has index one. Pell-type equations and continued fractions can establish this. Merely displaying one spectacular norm-one solution gives neither a minimality proof nor a complete description of the unit lattice.

The two real embeddings of a quadratic unit have absolute values that are reciprocals. A norm-minus-one unit has conjugate of opposite sign, yet the logarithmic vector still uses absolute values and lies on the same line. Torsion signs disappear in the logarithmic map, which is why they must be retained separately. When computing units numerically, exact norm verification and an index or minimality argument should accompany approximate logarithms; a nearly zero floating-point norm residual is not a certificate of an algebraic identity.