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Let $K$ have $r$ real embeddings and $s$ conjugate pairs of complex embeddings. The Dirichlet unit theorem states thatwhere $\mu (K)$ is the finite cyclic group of roots of unity in $K$. Thus the unit rank isEquivalently, there are units $u_1,\dots ,u_m$ such that every unit has a unique expression

Let $f$ be the monic cubic minimal polynomial of the algebraic integer $\alpha$, let ${\alpha }_1=\alpha ,{\alpha }_2,{\alpha }_3$ be its conjugates, and let ${\beta }_1=\beta ,{\beta }_2$ be the two conjugates of a nonrational $\beta \in {\mathcal{O}}_K$. Because the degrees three and two are coprime, the fields are linearly disjoint, andis a nonzero integer.

AssumeFor large $H(\beta )$ this makes ${\beta }_1$ bounded. Since $\beta$ is a quadratic algebraic integer, the height-Mahler measure formula givesand therefore $|{\beta }_2|\asymp H(\beta {)}^2$. The two remaining factors with $j=1$ are bounded, while the three factors with $j=2$ are $O(|{\beta }_2|)$. Consequently

We now use the standard effective norm form consequence of Part (b). Forwhere $1,\omega$ is an integral basis of $K$, the effective norm-form height estimate supplies effective constants $A,C>0$, depending only on $\alpha$ and $K$, such thatfor every nonzero $z\in V\cap {\mathcal{O}}_L$. Its proof factors $(z)$, balances a generator using the Dirichlet unit theorem, and applies the Baker lower bound for a homogeneous linear form in logarithms to the linear relations defining $V$; the coprime degrees $3$ and $2$ exclude a unit-family degeneracy.

Apply this estimate to $z=\alpha -\beta$. Since $\beta =\alpha -z$, the height inequalities imply $H(\beta )\le 2H(\alpha )H(z)$. HenceChoose the effective value $\epsilon =1/(2A)$. The last inequality bounds $H(\beta )$ effectively. The rational integers $\beta \in \mathbb{Z}$ are already covered by the stronger degree-three Liouville approximation theorem, and the Northcott theorem leaves only finitely many remaining quadratic integers of bounded height. Taking the minimum ofover this effective finite set gives an effective $c>0$ and provesfor every $\beta \in {\mathcal{O}}_K$.

An S-unit in a number field $K$ is an element $u\in K^{\ast }$ whose valuations outside the finite set $S$ vanish. Equivalently, it is a unit of the ring of S-integers ${\mathcal{O}}_{K,S}$. The S-unit group is finitely generated by the Dirichlet unit theorem together with generators arising from the inverted primes.

For a number field $K$ and a finite set $S$ of finite primes, an S-unit is an element of $K^{\times }$ whose discrete valuation is zero at every prime outside $S$. Its group is the unit group of the ring obtained by allowing denominators at $S$. The Dirichlet unit theorem and the valuation homomorphism to ${\mathbb{Z}}^S$ show that this group is a finitely generated abelian group.

The unit $\epsilon =1+\sqrt{2}$ has norm minus one and is the least unit greater than one in the positive real embedding. Dirichlet unit theorem therefore gives