Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 166 / 3

For $\beta \in K$, the nonzero algebraic number $\gamma =\alpha -\beta$ has degree at mostThe basic height inequality givesThe Liouville height inequality therefore yieldsThus one may take the explicit constant

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A suitable theorem is the Baker lower bound for a homogeneous linear form in logarithms. Let ${\alpha }_1,\dots ,{\alpha }_n$ be nonzero algebraic numbers with chosen logarithms and letChoose $A_i\ge 3$ to bound the degree-normalized Absolute logarithmic Weil height of ${\alpha }_i$ and $|\log {\alpha }_i|$, and put $B=\max (3,|b_1|,\dots ,|b_n|)$. If $\Lambda \ne 0$, thenwhere $C$ is an effectively computable constant depending only on $n$ and the degree of the number field generated by the ${\alpha }_i$.

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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 $\beta \in {\mathcal{O}}_K$.

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