S-unit

<h2 id="definition">Definition</h2>
Let K be a <a href="/facts/Number_field/e0K0TXSh">number field</a> with ring of integers R. Let S be a finite set of <a href="/facts/Prime_ideal/pmrGf6DA">prime ideals</a> of R. An element x of K is an S-unit if the <a href="/facts/Fractional_ideal/I5rvJ6nt">principal fractional ideal</a> (x) is a product of primes in S (to positive or negative powers). For the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> of <a href="/facts/Rational_integer/PUwwrawx">rational integers</a> Z one may take S to be a finite set of <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> and define an S-unit to be a <a href="/facts/Rational_number/VJQmyY05">rational number</a> whose numerator and denominator are <a href="/facts/Divisible/DKNa2GvC">divisible</a> only by the primes in S.

<h2 id="properties">Properties</h2>
The S-units form a multiplicative <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> containing the units of R. 
<a href="/facts/Dirichlet%27s_unit_theorem/IQDk3QAR">Dirichlet's unit theorem</a> holds for S-units: the group of S-units is <a href="/facts/Finitely_generated_abelian_group/dwtl7mp4">finitely generated</a>, with <a href="/facts/Rank_of_an_abelian_group/zo0y8Mj4">rank</a> (maximal number of multiplicatively independent elements) equal to r + s, where r is the rank of the unit group and s = |S|.

<h2 id="s-unit-equation">S-unit equation</h2>
The S-unit equation is a <a href="/facts/Diophantine_equation/mYaHp7oX">Diophantine equation</a>

u + v = 1
with u and v restricted to being S-units of K (or more generally, elements of a finitely generated subgroup of the multiplicative group of any field of characteristic zero). The number of solutions of this equation is finite<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> and the solutions are effectively determined using estimates for <a href="/facts/Linear_forms_in_logarithms/GXcd8RbV">linear forms in logarithms</a> as developed in <a href="/facts/Transcendental_number_theory/tjyLHm5G">transcendental number theory</a>. A variety of Diophantine equations are reducible in principle to some form of the S-unit equation: a notable example is <a href="/facts/Siegel%27s_theorem/J2AGT4ew">Siegel's theorem</a> on integral points on <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curves</a>, and more generally <a href="/facts/Superelliptic_curve/yeC2hSvN">superelliptic curves</a> of the form yn = f(x).
A computational solver for S-unit equation is available in the software <a href="/facts/SageMath/5mzh5TlB">SageMath</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<ul><li>Everest, Graham; <a href="/facts/Alfred_van_der_Poorten/XGqLmWnv">van der Poorten, Alf</a>; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. <a href="/facts/Providence,_RI/77kh9zJT">Providence, RI</a>: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. pp. 19–22. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-3387-1. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1033.11006">1033.11006</a>.</li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1978). Elliptic curves: Diophantine analysis. Grundlehren der mathematischen Wissenschaften. Vol. 231. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. pp. 128–153. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-08489-4.</li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1986). Algebraic number theory. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-94225-4. Chap. V.</li>
<li>Smart, Nigel (1998). <a href="https://archive.org/details/algorithmicresol0000smar/page/">The algorithmic resolution of Diophantine equations</a>. London Mathematical Society Student Texts. Vol. 41. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="https://archive.org/details/algorithmicresol0000smar/page/">Chap. 9</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-64156-X.</li>
<li><a href="/facts/J%C3%BCrgen_Neukirch/nzVc4G3F">Neukirch, Jürgen</a> (1986). Class field theory. Grundlehren der mathematischen Wissenschaften. Vol. 280. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. pp. 72–73. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-15251-2.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Alan_Baker_(mathematician)/1BAzUSML">Baker, Alan</a>; <a href="/facts/Gisbert_W%C3%BCstholz/FSPgk0lN">Wüstholz, Gisbert</a> (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. Vol. 9. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-88268-2.</li>
<li><a href="/facts/Enrico_Bombieri/MU6xU5vr">Bombieri, Enrico</a>; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-71229-3. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1130.11034">1130.11034</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Beukers, F.; Schlickewei, H. (1996). "The equation x+y=1 in finitely generated groups". Acta Arithmetica. 78 (2): 189–199. doi:10.4064/aa-78-2-189-199. ISSN 0065-1036. <a href="http://www.impan.pl/get/doi/10.4064/aa-78-2-189-199" target="_blank">http://www.impan.pl/get/doi/10.4064/aa-78-2-189-199</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Solve S-unit equation x + y = 1 — Sage Reference Manual v8.7: Algebraic Numbers and Number Fields". doc.sagemath.org. Retrieved 2019-04-16. <a href="http://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/S_unit_solver.html" target="_blank">http://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/S_unit_solver.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

S-unit open-in-new

S-unit