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Quadratic field
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<h2 id="ring-of-integers">Ring of integers</h2> <p class="note">Main article: <a href="/facts/Quadratic_integer/uGsLh5kL">Quadratic integer</a></p> <h2 id="discriminant">Discriminant</h2> <p>For a nonzero square free integer d {\displaystyle d} , the <a href="/facts/Discriminant_of_an_algebraic_number_field/mvv6MWKj">discriminant</a> of the quadratic field K = Q ( d ) {\displaystyle K=\mathbf {Q} ({\sqrt {d}})} is d {\displaystyle d} if d {\displaystyle d} is congruent to 1 {\displaystyle 1} modulo 4 {\displaystyle 4} , and otherwise 4 d {\displaystyle 4d} . For example, if d {\displaystyle d} is − 1 {\displaystyle -1} , then K {\displaystyle K} is the field of <a href="/facts/Gaussian_rational/X0jN1MnI">Gaussian rationals</a> and the discriminant is − 4 {\displaystyle -4} . The reason for such a distinction is that the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> of K {\displaystyle K} is generated by ( 1 + d ) / 2 {\displaystyle (1+{\sqrt {d}})/2} in the first case and by d {\displaystyle {\sqrt {d}}} in the second case. </p><p>The set of discriminants of quadratic fields is exactly the set of <a href="/facts/Fundamental_discriminant/yyrXDz4U">fundamental discriminants</a> (apart from 1 {\displaystyle 1} , which is a fundamental discriminant but not the discriminant of a quadratic field). </p> <h2 id="prime-factorization-into-ideals">Prime factorization into ideals</h2> <p>Any prime number p {\displaystyle p} gives rise to an ideal p O K {\displaystyle p{\mathcal {O}}_{K}} in the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> O K {\displaystyle {\mathcal {O}}_{K}} of a quadratic field K {\displaystyle K} . In line with general theory of <a href="/facts/Splitting_of_prime_ideals_in_Galois_extensions/G9I3GMKe">splitting of prime ideals in Galois extensions</a>, this may be<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> p {\displaystyle p} is inert ( p ) {\displaystyle (p)} is a prime ideal. The quotient ring is the <a href="/facts/Finite_field/UkMGJo3m">finite field</a> with p 2 {\displaystyle p^{2}} elements: O K / p O K = F p 2 {\displaystyle {\mathcal {O}}_{K}/p{\mathcal {O}}_{K}=\mathbf {F} _{p^{2}}} . p {\displaystyle p} splits ( p ) {\displaystyle (p)} is a product of two distinct prime ideals of O K {\displaystyle {\mathcal {O}}_{K}} . The quotient ring is the product O K / p O K = F p × F p {\displaystyle {\mathcal {O}}_{K}/p{\mathcal {O}}_{K}=\mathbf {F} _{p}\times \mathbf {F} _{p}} . p {\displaystyle p} is ramified ( p ) {\displaystyle (p)} is the square of a prime ideal of O K {\displaystyle {\mathcal {O}}_{K}} . The quotient ring contains non-zero <a href="/facts/Nilpotent/rERm2Pp7">nilpotent</a> elements. <p>The third case happens if and only if p {\displaystyle p} divides the discriminant D {\displaystyle D} . The first and second cases occur when the <a href="/facts/Kronecker_symbol/Mt7k50D0">Kronecker symbol</a> ( D / p ) {\displaystyle (D/p)} equals − 1 {\displaystyle -1} and + 1 {\displaystyle +1} , respectively. For example, if p {\displaystyle p} is an odd prime not dividing D {\displaystyle D} , then p {\displaystyle p} splits if and only if D {\displaystyle D} is congruent to a square modulo p {\displaystyle p} . The first two cases are, in a certain sense, equally likely to occur as p {\displaystyle p} runs through the primes—see <a href="/facts/Chebotarev_density_theorem/Ekg3wmbs">Chebotarev density theorem</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>The law of <a href="/facts/Quadratic_reciprocity/leCHbu5B">quadratic reciprocity</a> implies that the splitting behaviour of a prime p {\displaystyle p} in a quadratic field depends only on p {\displaystyle p} modulo D {\displaystyle D} , where D {\displaystyle D} is the field discriminant. </p> <h2 id="class-group">Class group</h2> <p>Determining the class group of a quadratic field extension can be accomplished using <a href="/facts/Minkowski%27s_bound/etVcd5RW">Minkowski's bound</a> and the <a href="/facts/Kronecker_symbol/Mt7k50D0">Kronecker symbol</a> because of the finiteness of the <a href="/facts/Ideal_class_group/LLn5Pwj1">class group</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> A quadratic field K = Q ( d ) {\displaystyle K=\mathbf {Q} ({\sqrt {d}})} has <a href="/facts/Discriminant_of_an_algebraic_number_field/mvv6MWKj">discriminant</a> Δ K = { d d ≡ 1 ( mod 4 ) 4 d d ≡ 2 , 3 ( mod 4 ) ; {\displaystyle \Delta _{K}={\begin{cases}d&d\equiv 1{\pmod {4}}\\4d&d\equiv 2,3{\pmod {4}};\end{cases}}} so the Minkowski bound is<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> M K = { 2 | Δ | / π d < 0 | Δ | / 2 d > 0. {\displaystyle M_{K}={\begin{cases}2{\sqrt {|\Delta |}}/\pi &d<0\\{\sqrt {|\Delta |}}/2&d>0.\end{cases}}} </p><p>Then, the ideal class group is generated by the prime ideals whose norm is less than M K {\displaystyle M_{K}} . This can be done by looking at the decomposition of the ideals ( p ) {\displaystyle (p)} for p ∈ Z {\displaystyle p\in \mathbf {Z} } prime where | p | < M k . {\displaystyle |p|<M_{k}.} <a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> page 72 These decompositions can be found using the <a href="/facts/Dedekind%E2%80%93Kummer_theorem/upkv7Bw9">Dedekind–Kummer theorem</a>. </p> <h2 id="quadratic-subfields-of-cyclotomic-fields">Quadratic subfields of cyclotomic fields</h2> <h3>The quadratic subfield of the prime cyclotomic field</h3> <p>A classical example of the construction of a quadratic field is to take the unique quadratic field inside the <a href="/facts/Cyclotomic_field/cGxMMEoR">cyclotomic field</a> generated by a primitive p {\displaystyle p} th root of unity, with p {\displaystyle p} an odd prime number. The uniqueness is a consequence of <a href="/facts/Galois_theory/54wscJfj">Galois theory</a>, there being a unique subgroup of <a href="/facts/Index_of_a_subgroup/ewY2A71Y">index</a> 2 {\displaystyle 2} in the Galois group over Q {\displaystyle \mathbf {Q} } . As explained at <a href="/facts/Gaussian_period/nx3pvsNF">Gaussian period</a>, the discriminant of the quadratic field is p {\displaystyle p} for p = 4 n + 1 {\displaystyle p=4n+1} and − p {\displaystyle -p} for p = 4 n + 3 {\displaystyle p=4n+3} . This can also be predicted from enough <a href="/facts/Ramification_(mathematics)/y4mDBqjo">ramification</a> theory. In fact, p {\displaystyle p} is the only prime that ramifies in the cyclotomic field, so p {\displaystyle p} is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants − 4 p {\displaystyle -4p} and 4 p {\displaystyle 4p} in the respective cases. </p> <h3>Other cyclotomic fields</h3> <p>If one takes the other cyclotomic fields, they have Galois groups with extra 2 {\displaystyle 2} -torsion, so contain at least three quadratic fields. In general a quadratic field of field discriminant D {\displaystyle D} can be obtained as a subfield of a cyclotomic field of D {\displaystyle D} -th roots of unity. This expresses the fact that the <a href="/facts/Conductor_(class_field_theory)/KFN786uM">conductor</a> of a quadratic field is the absolute value of its discriminant, a special case of the <a href="/facts/Conductor-discriminant_formula/N12Splkt">conductor-discriminant formula</a>. </p> <h2 id="orders-of-quadratic-number-fields-of-small-discriminant">Orders of quadratic number fields of small discriminant</h2> <p>The following table shows some <a href="/facts/Order_(ring_theory)/4JzIeDKz">orders</a> of small discriminant of quadratic fields. The <i>maximal order</i> of an algebraic number field is its <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a>, and the discriminant of the maximal order is the discriminant of the field. The discriminant of a non-maximal order is the product of the discriminant of the corresponding maximal order by the square of the determinant of the matrix that expresses a basis of the non-maximal order over a basis of the maximal order. All these discriminants may be defined by the formula of <a href="/facts/Discriminant_of_an_algebraic_number_field/mvv6MWKj">Discriminant of an algebraic number field § Definition</a>. </p><p>For real quadratic integer rings, the <a href="/facts/Ideal_class_group/LLn5Pwj1">ideal class number</a>, which measures the failure of unique factorization, is given in <a href="https://oeis.org/A003649">OEIS A003649</a>; for the imaginary case, they are given in <a href="https://oeis.org/A000924">OEIS A000924</a>. </p> <table><tbody><tr><th>Order</th><th>Discriminant</th><th>Class number</th><th>Units</th><th>Comments</th></tr><tr><td> Z [ − 5 ] {\displaystyle \mathbf {Z} \left[{\sqrt {-5}}\right]} </td><td> − 20 {\displaystyle -20} </td><td> 2 {\displaystyle 2} </td><td> ± 1 {\displaystyle \pm 1} </td><td>Ideal classes ( 1 ) {\displaystyle (1)} , ( 2 , 1 + − 5 ) {\displaystyle (2,1+{\sqrt {-5}})} </td></tr><tr><td> Z [ ( 1 + − 19 ) / 2 ] {\displaystyle \mathbf {Z} \left[(1+{\sqrt {-19}})/2\right]} </td><td> − 19 {\displaystyle -19} </td><td> 1 {\displaystyle 1} </td><td> ± 1 {\displaystyle \pm 1} </td><td><a href="/facts/Principal_ideal_domain/JCwAUnyW">Principal ideal domain</a>, not <a href="/facts/Euclidean_domain/ydwCD0fz">Euclidean</a></td></tr><tr><td> Z [ 2 − 1 ] {\displaystyle \mathbf {Z} \left[2{\sqrt {-1}}\right]} </td><td> − 16 {\displaystyle -16} </td><td> 1 {\displaystyle 1} </td><td> ± 1 {\displaystyle \pm 1} </td><td>Non-maximal order</td></tr><tr><td> Z [ ( 1 + − 15 ) / 2 ] {\displaystyle \mathbf {Z} \left[(1+{\sqrt {-15}})/2\right]} </td><td> − 15 {\displaystyle -15} </td><td> 2 {\displaystyle 2} </td><td> ± 1 {\displaystyle \pm 1} </td><td>Ideal classes ( 1 ) {\displaystyle (1)} , ( 1 , ( 1 + − 15 ) / 2 ) {\displaystyle (1,(1+{\sqrt {-15}})/2)} </td></tr><tr><td> Z [ − 3 ] {\displaystyle \mathbf {Z} \left[{\sqrt {-3}}\right]} </td><td> − 12 {\displaystyle -12} </td><td> 1 {\displaystyle 1} </td><td> ± 1 {\displaystyle \pm 1} </td><td>Non-maximal order</td></tr><tr><td> Z [ ( 1 + − 11 ) / 2 ] {\displaystyle \mathbf {Z} \left[(1+{\sqrt {-11}})/2\right]} </td><td> − 11 {\displaystyle -11} </td><td> 1 {\displaystyle 1} </td><td> ± 1 {\displaystyle \pm 1} </td><td>Euclidean</td></tr><tr><td> Z [ − 2 ] {\displaystyle \mathbf {Z} \left[{\sqrt {-2}}\right]} </td><td> − 8 {\displaystyle -8} </td><td> 1 {\displaystyle 1} </td><td> ± 1 {\displaystyle \pm 1} </td><td>Euclidean</td></tr><tr><td> Z [ ( 1 + − 7 ) / 2 ] {\displaystyle \mathbf {Z} \left[(1+{\sqrt {-7}})/2\right]} </td><td> − 7 {\displaystyle -7} </td><td> 1 {\displaystyle 1} </td><td> ± 1 {\displaystyle \pm 1} </td><td><a href="/facts/Kleinian_integers/nxn45NR0">Kleinian integers</a></td></tr><tr><td> Z [ − 1 ] {\displaystyle \mathbf {Z} \left[{\sqrt {-1}}\right]} </td><td> − 4 {\displaystyle -4} </td><td> 1 {\displaystyle 1} </td><td> ± 1 , ± i {\displaystyle \pm 1,\pm i} (cyclic of order 4 {\displaystyle 4} )</td><td><a href="/facts/Gaussian_integers/zj82AhtL">Gaussian integers</a></td></tr><tr><td> Z [ ( 1 + − 3 ) / 2 ] {\displaystyle \mathbf {Z} \left[(1+{\sqrt {-3}})/2\right]} </td><td> − 3 {\displaystyle -3} </td><td> 1 {\displaystyle 1} </td><td> ± 1 , ( ± 1 ± − 3 ) / 2 {\displaystyle \pm 1,(\pm 1\pm {\sqrt {-3}})/2} .</td><td><a href="/facts/Eisenstein_integers/UGQmFI8c">Eisenstein integers</a></td></tr><tr><td> Z [ − 21 ] {\displaystyle \mathbf {Z} \left[{\sqrt {-21}}\right]} </td><td> − 84 {\displaystyle -84} </td><td> 4 {\displaystyle 4} </td><td></td><td>Class group non-cyclic: ( Z / 2 Z ) 2 {\displaystyle (\mathbf {Z} /2\mathbf {Z} )^{2}} </td></tr><tr><td> Z [ ( 1 + 5 ) / 2 ] {\displaystyle \mathbf {Z} \left[(1+{\sqrt {5}})/2\right]} </td><td> 5 {\displaystyle 5} </td><td> 1 {\displaystyle 1} </td><td> ± ( ( 1 + 5 ) / 2 ) n {\displaystyle \pm ((1+{\sqrt {5}})/2)^{n}} (norm ( − 1 ) n {\displaystyle (-1)^{n}} )</td><td></td></tr><tr><td> Z [ 2 ] {\displaystyle \mathbf {Z} \left[{\sqrt {2}}\right]} </td><td> 8 {\displaystyle 8} </td><td> 1 {\displaystyle 1} </td><td> ± ( 1 + 2 ) n {\displaystyle \pm (1+{\sqrt {2}})^{n}} (norm ( − 1 ) n {\displaystyle (-1)^{n}} )</td><td></td></tr><tr><td> Z [ 3 ] {\displaystyle \mathbf {Z} \left[{\sqrt {3}}\right]} </td><td> 12 {\displaystyle 12} </td><td> 1 {\displaystyle 1} </td><td> ± ( 2 + 3 ) n {\displaystyle \pm (2+{\sqrt {3}})^{n}} (norm 1 {\displaystyle 1} )</td><td></td></tr><tr><td> Z [ ( 1 + 13 ) / 2 ] {\displaystyle \mathbf {Z} \left[(1+{\sqrt {13}})/2\right]} </td><td> 13 {\displaystyle 13} </td><td> 1 {\displaystyle 1} </td><td> ± ( ( 3 + 13 ) / 2 ) n {\displaystyle \pm ((3+{\sqrt {13}})/2)^{n}} (norm ( − 1 ) n {\displaystyle (-1)^{n}} )</td><td></td></tr><tr><td> Z [ ( 1 + 17 ) / 2 ] {\displaystyle \mathbf {Z} \left[(1+{\sqrt {17}})/2\right]} </td><td> 17 {\displaystyle 17} </td><td> 1 {\displaystyle 1} </td><td> ± ( 4 + 17 ) n {\displaystyle \pm (4+{\sqrt {17}})^{n}} (norm ( − 1 ) n {\displaystyle (-1)^{n}} )</td><td></td></tr><tr><td> Z [ 5 ] {\displaystyle \mathbf {Z} \left[{\sqrt {5}}\right]} </td><td> 20 {\displaystyle 20} </td><td> 1 {\displaystyle 1} </td><td> ± ( 5 + 2 ) n {\displaystyle \pm ({\sqrt {5}}+2)^{n}} (norm ( − 1 ) n {\displaystyle (-1)^{n}} )</td><td>Non-maximal order</td></tr></tbody></table> <p>Some of these examples are listed in Artin, <i>Algebra</i> (2nd ed.), §13.8. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Eisenstein%E2%80%93Kronecker_number/3y688ImY">Eisenstein–Kronecker number</a></li> <li><a href="/facts/Genus_character/YcfXMeXE">Genus character</a></li> <li><a href="/facts/Heegner_number/ExPvl9UM">Heegner number</a></li> <li><a href="/facts/Infrastructure_(number_theory)/eVE0scfM">Infrastructure (number theory)</a></li> <li><a href="/facts/Quadratic_integer/uGsLh5kL">Quadratic integer</a></li> <li><a href="/facts/Quadratic_irrational/QXKP2PnT">Quadratic irrational</a></li> <li><a href="/facts/Stark%E2%80%93Heegner_theorem/o7oDA25G">Stark–Heegner theorem</a></li> <li><a href="/facts/Dedekind_zeta_function/Nj8PwFo8">Dedekind zeta function</a></li> <li><a href="/facts/Quadratically_closed_field/LswBg2U4">Quadratically closed field</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Buell, Duncan (1989), <a href="https://archive.org/details/binaryquadraticf0000buel"><i>Binary quadratic forms: classical theory and modern computations</i></a>, <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-97037-1 Chapter 6.</li> <li><a href="/facts/Pierre_Samuel/r63UjoTx">Samuel, Pierre</a> (1972), <i>Algebraic Theory of Numbers</i> (Hardcover ed.), Paris / Boston: Hermann / Houghton Mifflin Company, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-901-66506-5 <ul><li>Samuel, Pierre (2008), <a href="https://books.google.com/books?id=u9vCAgAAQBAJ"><i>Algebraic Theory of Numbers</i></a> (Paperback ed.), Dover, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-46666-8</li></ul></li> <li><a href="/facts/Ian_Stewart_(mathematician)/HxAST88r">Stewart, I. N.</a>; Tall, D. O. (1979), <i>Algebraic number theory</i>, Chapman and Hall, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-412-13840-9 Chapter 3.1.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/QuadraticField.html">"Quadratic Field"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="https://www.encyclopediaofmath.org/index.php?title=Quadratic_field">"Quadratic field"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Stevenhagen. "Number Rings" (PDF). p. 36. <a href="http://websites.math.leidenuniv.nl/algebra/ant.pdf" target="_blank">http://websites.math.leidenuniv.nl/algebra/ant.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Samuel 1972, pp. 76f - Samuel, Pierre (1972), Algebraic Theory of Numbers (Hardcover ed.), Paris / Boston: Hermann / Houghton Mifflin Company, ISBN 978-0-901-66506-5 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Stein, William. "Algebraic Number Theory, A Computational Approach" (PDF). pp. 77–86. <a href="https://wstein.org/books/ant/ant.pdf" target="_blank">https://wstein.org/books/ant/ant.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Conrad, Keith. "CLASS GROUP CALCULATIONS" (PDF). <a href="https://kconrad.math.uconn.edu/blurbs/gradnumthy/classgpex.pdf" target="_blank">https://kconrad.math.uconn.edu/blurbs/gradnumthy/classgpex.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Stevenhagen. "Number Rings" (PDF). p. 36. <a href="http://websites.math.leidenuniv.nl/algebra/ant.pdf" target="_blank">http://websites.math.leidenuniv.nl/algebra/ant.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>