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Glossary of arithmetic and diophantine geometry
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<p> This is a glossary of arithmetic and diophantine geometry in <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, areas growing out of the traditional study of <a href="/facts/Diophantine_equation/mYaHp7oX">Diophantine equations</a> to encompass large parts of <a href="/facts/Number_theory/zhoa7zrc">number theory</a> and <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>. Much of the theory is in the form of proposed <a href="/facts/Conjecture/oPWUb7SF">conjectures</a>, which can be related at various levels of generality. </p><p><a href="/facts/Diophantine_geometry/H2x4wzUE">Diophantine geometry</a> in general is the study of <a href="/facts/Algebraic_varieties/Vk7f97vn">algebraic varieties</a> <i>V</i> over fields <i>K</i> that are finitely generated over their <a href="/facts/Prime_field/D2VzcQaG">prime fields</a>—including as of special interest <a href="/facts/Number_field/e0K0TXSh">number fields</a> and <a href="/facts/Finite_field/UkMGJo3m">finite fields</a>—and over <a href="/facts/Local_field/vqIgPIX3">local fields</a>. Of those, only the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> are <a href="/facts/Algebraically_closed/HuVQClok">algebraically closed</a>; over any other <i>K</i> the existence of points of <i>V</i> with coordinates in <i>K</i> is something to be proved and studied as an extra topic, even knowing the geometry of <i>V</i>. </p><p><a href="/facts/Arithmetic_geometry/Bs1mfNu4">Arithmetic geometry</a> can be more generally defined as the study of <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a> of finite type over the <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">spectrum</a> of the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a>. Arithmetic geometry has also been defined as the application of the techniques of algebraic geometry to problems in <a href="/facts/Number_theory/zhoa7zrc">number theory</a>. </p><p>See also the glossary of number theory terms at <a href="/facts/Glossary_of_number_theory/fk4t8vG3">Glossary of number theory</a>. </p>