Isotropic quadratic form

<h2 id="hyperbolic-plane">Hyperbolic plane</h2>
Not to be confused with the <a href="/facts/Plane_(geometry)/tAUtRFcn">plane</a> in <a href="/facts/Hyperbolic_geometry/DjTMKdgy">hyperbolic geometry</a>.
Let F be a field of <a href="/facts/Characteristic_(field)/D2VzcQaG">characteristic</a> not 2 and V = F2. If we consider the general element (x, y) of V, then the quadratic forms q = xy and r = x2 − y2 are equivalent since there is a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> on V that makes q look like r, and vice versa. Evidently, (V, q) and (V, r) are isotropic. This example is called the hyperbolic plane in the theory of <a href="/facts/Quadratic_form/l2n1jXPe">quadratic forms</a>. A common instance has F = <a href="/facts/Real_number/R02gw5Pb">real numbers</a> in which case {x ∈ V : q(x) = nonzero constant} and {x ∈ V : r(x) = nonzero constant} are <a href="/facts/Hyperbola/6R6ER0mY">hyperbolas</a>. In particular, {x ∈ V : r(x) = 1} is the <a href="/facts/Unit_hyperbola/AvwG6gJ9">unit hyperbola</a>. The notation ⟨1⟩ ⊕ ⟨−1⟩ has been used by Milnor and Husemoller<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>: 9  for the hyperbolic plane as the signs of the terms of the <a href="/facts/Polynomial/Lzak8VVx">bivariate polynomial</a> r are exhibited.
The affine hyperbolic plane was described by <a href="/facts/Emil_Artin/rI3vIFo2">Emil Artin</a> as a quadratic space with basis {M, N} satisfying M2 = N2 = 0, NM = 1, where the products represent the quadratic form.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
Through the <a href="/facts/Polarization_identity/l3Vr7sC7">polarization identity</a> the quadratic form is related to a <a href="/facts/Symmetric_bilinear_form/UI9b6RK7">symmetric bilinear form</a> B(u, v) = ⁠1/4⁠(q(u + v) − q(u − v)).
Two vectors u and v are <a href="/facts/Orthogonal/JVIjYPog">orthogonal</a> when B(u, v) = 0. In the case of the hyperbolic plane, such u and v are <a href="/facts/Hyperbolic-orthogonal/webbXEhz">hyperbolic-orthogonal</a>.

<h2 id="split-quadratic-space">Split quadratic space</h2>
A space with quadratic form is split (or metabolic) if there is a subspace which is equal to its own <a href="/facts/Orthogonal_complement/a13oQBsz">orthogonal complement</a>; equivalently, the index of isotropy is equal to half the dimension.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>: 57  The hyperbolic plane is an example, and over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>: 12, 3 

<h2 id="relation-with-classification-of-quadratic-forms">Relation with classification of quadratic forms</h2>
From the point of view of classification of quadratic forms, spaces with definite quadratic forms are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field F, classification of definite quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. By <a href="/facts/Witt%27s_decomposition_theorem/zpW1w6Il">Witt's decomposition theorem</a>, every <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> over a field is an <a href="/facts/Orthogonal_direct_sum/Y6PBJnjg">orthogonal direct sum</a> of a split space and a space with definite quadratic form.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>: 56 

<h2 id="field-theory">Field theory</h2>
<ul><li>If F is an <a href="/facts/Algebraically_closed/HuVQClok">algebraically closed</a> field, for example, the field of <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a>, and (V, q) is a quadratic space of dimension at least two, then it is isotropic.</li>
<li>If F is a <a href="/facts/Finite_field/UkMGJo3m">finite field</a> and (V, q) is a quadratic space of dimension at least three, then it is isotropic (this is a consequence of the <a href="/facts/Chevalley%E2%80%93Warning_theorem/xjbRnJwq">Chevalley–Warning theorem</a>).</li>
<li>If F is the field Qp of <a href="/facts/P-adic_number/SDgDbkLF">p-adic numbers</a> and (V, q) is a quadratic space of dimension at least five, then it is isotropic.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Isotropic_line/eMeRF6mB">Isotropic line</a></li>
<li><a href="/facts/Polar_space/EF8Dyqqp">Polar space</a></li>
<li><a href="/facts/Witt_group/bvjfYwRr">Witt group</a></li>
<li><a href="/facts/Witt_ring_(forms)/bvjfYwRr">Witt ring (forms)</a></li>
<li><a href="/facts/Universal_quadratic_form/LQmz8I8F">Universal quadratic form</a></li></ul>

<ul><li>Pete L. Clark, <a href="http://www.math.miami.edu/~armstrong/685fa12/pete_clark.pdf">Quadratic forms chapter I: Witts theory</a> from <a href="/facts/University_of_Miami/2rPFYefT">University of Miami</a> in <a href="/facts/Coral_Gables,_Florida/CcMbTzfg">Coral Gables, Florida</a>.</li>
<li><a href="/facts/Tsit_Yuen_Lam/S1ewInkE">Tsit Yuen Lam</a> (1973) Algebraic Theory of Quadratic Forms, §1.3 Hyperbolic plane and hyperbolic spaces, <a href="/facts/W._A._Benjamin/qDQI0oh4">W. A. Benjamin</a>.</li>
<li>Tsit Yuen Lam (2005) Introduction to Quadratic Forms over Fields, <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a> <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1095-2 .</li>
<li><a href="/facts/O._Timothy_O%27Meara/4ST6KEXf">O'Meara, O.T</a> (1963). Introduction to Quadratic Forms. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. p. 94 §42D Isotropy. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-66564-1.</li>
<li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (2000) [1973]. <a href="https://archive.org/details/courseinarithmet00serr">A Course in Arithmetic</a>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>: Classics in mathematics. Vol. 7 (reprint of 3rd ed.). <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90040-3. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1034.11003">1034.11003</a>.</li></ul>

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

<ol>
<li id="fn:1">Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016. <a href="3-540-06009-X" target="_blank">3-540-06009-X</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016. <a href="3-540-06009-X" target="_blank">3-540-06009-X</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Emil Artin (1957) Geometric Algebra, page 119 via Internet Archive <a href="/wiki/Emil_Artin" target="_blank">/wiki/Emil_Artin</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016. <a href="3-540-06009-X" target="_blank">3-540-06009-X</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016. <a href="3-540-06009-X" target="_blank">3-540-06009-X</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016. <a href="3-540-06009-X" target="_blank">3-540-06009-X</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Isotropic quadratic form open-in-new

Isotropic quadratic form