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Glossary of field theory
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<h2 id="definition-of-a-field">Definition of a field</h2> <p>A field is a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> (<i>F</i>, +, *) in which 0 ≠ 1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division. </p><p>The non-zero elements of a field <i>F</i> form an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> under multiplication; this group is typically denoted by <i>F</i>×; </p><p>The <a href="/facts/Polynomial_ring/ua3vk8Ih">ring of polynomials</a> in the variable <i>x</i> with coefficients in <i>F</i> is denoted by <i>F</i>[<i>x</i>]. </p> <h2 id="basic-definitions">Basic definitions</h2> <a href="/facts/Characteristic_(algebra)/D2VzcQaG">Characteristic</a> The <i>characteristic</i> of the field <i>F</i> is the smallest positive <a href="/facts/Integer/PUwwrawx">integer</a> <i>n</i> such that <i>n</i>·1 = 0; here <i>n</i>·1 stands for <i>n</i> summands 1 + 1 + 1 + ... + 1. If no such <i>n</i> exists, we say the characteristic is zero. Every non-zero characteristic is a <a href="/facts/Prime_number/L20FLkgn">prime number</a>. For example, the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>, the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> and the <a href="/facts/P-adic_numbers/SDgDbkLF"><i>p</i>-adic numbers</a> have characteristic 0, while the finite field Z<i>p</i> with <i>p</i> being prime has characteristic <i>p</i>. Subfield A <i>subfield</i> of a field <i>F</i> is a <a href="/facts/Subset/SJGERmbA">subset</a> of <i>F</i> which is closed under the field operation + and * of <i>F</i> and which, with these operations, forms itself a field. <a href="/facts/Prime_field/D2VzcQaG">Prime field</a> The <i>prime field</i> of the field <i>F</i> is the unique smallest subfield of <i>F</i>. <a href="/facts/Field_extension/L7yIDtyK">Extension field</a> If <i>F</i> is a subfield of <i>E</i> then <i>E</i> is an <i>extension field</i> of <i>F</i>. We then also say that <i>E</i>/<i>F</i> is a <i>field extension</i>. <a href="/facts/Degree_of_a_field_extension/hY5LgHYk">Degree of an extension</a> Given an extension <i>E</i>/<i>F</i>, the field <i>E</i> can be considered as a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over the field <i>F</i>, and the <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimension</a> of this vector space is the <i>degree</i> of the extension, denoted by [<i>E</i> : <i>F</i>]. Finite extension A <i>finite extension</i> is a field extension whose degree is finite. <a href="/facts/Algebraic_extension/GAs76USY">Algebraic extension</a> If an element <i>α</i> of an extension field <i>E</i> over <i>F</i> is the <a href="/facts/Root/x2U60Gdm">root</a> of a non-zero polynomial in <i>F</i>[<i>x</i>], then <i>α</i> is <i>algebraic</i> over <i>F</i>. If every element of <i>E</i> is algebraic over <i>F</i>, then <i>E</i>/<i>F</i> is an <i>algebraic extension</i>. Generating set Given a field extension <i>E</i>/<i>F</i> and a subset <i>S</i> of <i>E</i>, we write <i>F</i>(<i>S</i>) for the smallest subfield of <i>E</i> that contains both <i>F</i> and <i>S</i>. It consists of all the elements of <i>E</i> that can be obtained by repeatedly using the operations +, −, *, / on the elements of <i>F</i> and <i>S</i>. If <i>E</i> = <i>F</i>(<i>S</i>), we say that <i>E</i> is generated by <i>S</i> over <i>F</i>. <a href="/facts/Primitive_element_(field_theory)/xegn8Bpl">Primitive element</a> An element <i>α</i> of an extension field <i>E</i> over a field <i>F</i> is called a <i>primitive element</i> if <i>E</i>=<i>F</i>(<i>α</i>), the smallest extension field containing <i>α</i>. Such an extension is called a <a href="/facts/Simple_extension/xegn8Bpl">simple extension</a>. <a href="/facts/Splitting_field/m38eWUH1">Splitting field</a> A field extension generated by the complete factorisation of a polynomial. <a href="/facts/Normal_extension/HCIHmH4t">Normal extension</a> A field extension generated by the complete factorisation of a set of polynomials. <a href="/facts/Separable_extension/9KBvXmz8">Separable extension</a> An extension generated by roots of <a href="/facts/Separable_polynomial/5mcxhRGw">separable polynomials</a>. <a href="/facts/Perfect_field/0JJvr8Xx">Perfect field</a> A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields, are perfect. Imperfect degree Let <i>F</i> be a field of characteristic <i>p</i> > 0; then <i>F</i><i>p</i> is a subfield. The degree [<i>F</i> : <i>F</i><i>p</i>] is called the <i>imperfect degree</i> of <i>F</i>. The field <i>F</i> is perfect if and only if its imperfect degree is <i>1</i>. For example, if <i>F</i> is a function field of <i>n</i> variables over a finite field of characteristic <i>p</i> > 0, then its imperfect degree is <i>p</i><i>n</i>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> <a href="/facts/Algebraically_closed_field/HuVQClok">Algebraically closed field</a> A field <i>F</i> is <i>algebraically closed</i> if every polynomial in <i>F</i>[<i>x</i>] has a root in <i>F</i>; equivalently: every polynomial in <i>F</i>[<i>x</i>] is a product of linear factors. <a href="/facts/Algebraic_closure/yjewG1mM">Algebraic closure</a> An <i>algebraic closure</i> of a field <i>F</i> is an algebraic extension of <i>F</i> which is algebraically closed. Every field has an algebraic closure, and it is unique up to an isomorphism that fixes <i>F</i>. <a href="/facts/Transcendental_element/Bl87J5mB">Transcendental</a> Those elements of an extension field of <i>F</i> that are not algebraic over <i>F</i> are <i>transcendental</i> over <i>F</i>. Algebraically independent elements Elements of an extension field of <i>F</i> are <i>algebraically independent</i> over <i>F</i> if they don't satisfy any non-zero polynomial equation with coefficients in <i>F</i>. <a href="/facts/Transcendence_degree/ubxWrbjJ">Transcendence degree</a> The number of algebraically independent transcendental elements in a field extension. It is used to define the <a href="/facts/Dimension_of_an_algebraic_variety/sXqIHcYc">dimension of an algebraic variety</a>. <h2 id="homomorphisms">Homomorphisms</h2> Field homomorphism A <i>field homomorphism</i> between two fields <i>E</i> and <i>F</i> is a <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a>, i.e., a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> <i>f</i> : <i>E</i> → <i>F</i> such that, for all <i>x</i>, <i>y</i> in <i>E</i>, <i>f</i>(<i>x</i> + <i>y</i>) = <i>f</i>(<i>x</i>) + <i>f</i>(<i>y</i>) <i>f</i>(<i>xy</i>) = <i>f</i>(<i>x</i>) <i>f</i>(<i>y</i>) <i>f</i>(1) = 1. For fields <i>E</i> and <i>F</i>, these properties imply that <i>f</i>(0) = 0, <i>f</i>(<i>x</i>−1) = <i>f</i>(<i>x</i>)−1 for <i>x</i> in <i>E</i>×, and that <i>f</i> is <a href="/facts/Injective/5ES6tqf5">injective</a>. Fields, together with these homomorphisms, form a <a href="/facts/Category_theory/6zS3DXPa">category</a>. Two fields <i>E</i> and <i>F</i> are called isomorphic if there exists a <a href="/facts/Bijective/j4gTuTmW">bijective</a> homomorphism <i>f</i> : <i>E</i> → <i>F</i>. The two fields are then identical for all practical purposes; however, not necessarily in a <i>unique</i> way. See, for example, <i><a href="/facts/Complex_conjugate/7TEaoQDe">Complex conjugate</a></i>. <h2 id="types-of-fields">Types of fields</h2> <a href="/facts/Finite_field/UkMGJo3m">Finite field</a> A field with finitely many elements, a.k.a. Galois field. <a href="/facts/Ordered_field/an0prSgG">Ordered field</a> A field with a <a href="/facts/Total_order/xnTtmuYJ">total order</a> compatible with its operations. <a href="/facts/Rational_number/VJQmyY05">Rational numbers</a> <a href="/facts/Real_number/R02gw5Pb">Real numbers</a> <a href="/facts/Complex_number/2w7mqI7e">Complex numbers</a> <a href="/facts/Number_field/e0K0TXSh">Number field</a> Finite extension of the field of rational numbers. <a href="/facts/Algebraic_number/MkH1PzTK">Algebraic numbers</a> The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers. Their detailed properties are studied in <a href="/facts/Algebraic_number_theory/LPzDT5CA">algebraic number theory</a>. <a href="/facts/Quadratic_field/dUnSDN4e">Quadratic field</a> A degree-two extension of the rational numbers. <a href="/facts/Cyclotomic_field/cGxMMEoR">Cyclotomic field</a> An extension of the rational numbers generated by a <a href="/facts/Root_of_unity/EqH0ho1Y">root of unity</a>. <a href="/facts/Totally_real_field/4rulk9kd">Totally real field</a> A number field generated by a root of a polynomial, having all its roots real numbers. <a href="/facts/Formally_real_field/IXsKBxEs">Formally real field</a> <a href="/facts/Real_closed_field/Vj1Un2fw">Real closed field</a> <a href="/facts/Global_field/na458xWs">Global field</a> A number field or a function field of one variable over a finite field. <a href="/facts/Local_field/vqIgPIX3">Local field</a> A completion of some global field (<a href="/facts/W.r.t./gos7I1Bk">w.r.t.</a> a prime of the integer ring). <a href="/facts/Complete_field/g4H2LRRG">Complete field</a> A field complete w.r.t. to some valuation. <a href="/facts/Pseudo_algebraically_closed_field/I0IERqng">Pseudo algebraically closed field</a> A field in which every variety has a <a href="/facts/Rational_point/j5v4cxuE">rational point</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> <a href="/facts/Henselian_field/rJaECl4s">Henselian field</a> A field satisfying <a href="/facts/Hensel_lemma/CPCADITS">Hensel lemma</a> w.r.t. some valuation. A generalization of complete fields. <a href="/facts/Hilbertian_field/aVBYjqNe">Hilbertian field</a> A field satisfying <a href="/facts/Hilbert%27s_irreducibility_theorem/1GCqas32">Hilbert's irreducibility theorem</a>: formally, one for which the <a href="/facts/Projective_line/PhRgf87G">projective line</a> is not <a href="/facts/Thin_set_(Serre)/aVBYjqNe">thin in the sense of Serre</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> Kroneckerian field A totally real algebraic number field or a totally imaginary quadratic extension of a totally real field.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> <a href="/facts/CM-field/vL0Jcded">CM-field</a> or J-field An algebraic number field which is a totally imaginary quadratic extension of a totally real field.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> <a href="/facts/Linked_field/ANTLI9zK">Linked field</a> A field over which no <a href="/facts/Biquaternion_algebra/Lq8pQdTQ">biquaternion algebra</a> is a <a href="/facts/Division_algebra/xp5Bepx3">division algebra</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Frobenius field A <a href="/facts/Pseudo_algebraically_closed_field/I0IERqng">pseudo algebraically closed field</a> whose <a href="/facts/Absolute_Galois_group/fvcDwzWq">absolute Galois group</a> has the embedding property.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> <h2 id="field-extensions">Field extensions</h2> <p>Let <i>E</i>/<i>F</i> be a field extension. </p> <a href="/facts/Algebraic_extension/GAs76USY">Algebraic extension</a> An extension in which every element of <i>E</i> is algebraic over <i>F</i>. <a href="/facts/Simple_extension/xegn8Bpl">Simple extension</a> An extension which is generated by a single element, called a primitive element, or generating element.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> The <a href="/facts/Primitive_element_theorem/u5DvBoTl">primitive element theorem</a> classifies such extensions.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> <a href="/facts/Normal_extension/HCIHmH4t">Normal extension</a> An extension that splits a family of polynomials: every root of the minimal polynomial of an element of <i>E</i> over <i>F</i> is also in <i>E</i>. <a href="/facts/Separable_extension/9KBvXmz8">Separable extension</a> An algebraic extension in which the minimal polynomial of every element of <i>E</i> over <i>F</i> is a <a href="/facts/Separable_polynomial/5mcxhRGw">separable polynomial</a>, that is, has distinct roots.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> <a href="/facts/Galois_extension/52fzNQbv">Galois extension</a> A normal, separable field extension. <a href="/facts/Primary_extension/F2IJl0RI">Primary extension</a> An extension <i>E</i>/<i>F</i> such that the algebraic closure of <i>F</i> in <i>E</i> is <a href="/facts/Purely_inseparable/IoGzhxB7">purely inseparable</a> over <i>F</i>; equivalently, <i>E</i> is <a href="/facts/Linearly_disjoint/Totpn0fa">linearly disjoint</a> from the <a href="/facts/Separable_closure/yjewG1mM">separable closure</a> of <i>F</i>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> <a href="/facts/Purely_transcendental_extension/L7yIDtyK">Purely transcendental extension</a> An extension <i>E</i>/<i>F</i> in which every element of <i>E</i> not in <i>F</i> is transcendental over <i>F</i>.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> <a href="/facts/Regular_extension/IkoQC9xM">Regular extension</a> An extension <i>E</i>/<i>F</i> such that <i>E</i> is separable over <i>F</i> and <i>F</i> is algebraically closed in <i>E</i>.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> <a href="/facts/Simple_radical_extension/1wJhjwNs">Simple radical extension</a> A <a href="/facts/Simple_extension/xegn8Bpl">simple extension</a> <i>E</i>/<i>F</i> generated by a single element <i>α</i> satisfying <i>α</i><i>n</i> = <i>b</i> for an element <i>b</i> of <i>F</i>. In <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> <i>p</i>, we also take an extension by a root of an <a href="/facts/Artin%E2%80%93Schreier_theory/nC8hvw8k">Artin–Schreier polynomial</a> to be a simple radical extension.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> <a href="/facts/Radical_extension/1wJhjwNs">Radical extension</a> A tower <i>F</i> = <i>F</i>0 < <i>F</i>1 < ⋅⋅⋅ < <i>F</i><i>k</i> = <i>E</i> where each extension <i>F</i><i>i</i> / <i>F</i><i>i</i>−1 is a simple radical extension.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> <a href="/facts/Self-regular_extension/IkoQC9xM">Self-regular extension</a> An extension <i>E</i>/<i>F</i> such that <i>E</i> ⊗<i>F</i> <i>E</i> is an integral domain.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> Totally transcendental extension An extension <i>E</i>/<i>F</i> such that <i>F</i> is algebraically closed in <i>F</i>.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> Distinguished class A class <i>C</i> of field extensions with the three properties<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> <ol><li>If <i>E</i> is a C-extension of <i>F</i> and <i>F</i> is a C-extension of <i>K</i> then <i>E</i> is a C-extension of <i>K</i>.</li> <li>If <i>E</i> and <i>F</i> are C-extensions of <i>K</i> in a common overfield <i>M</i>, then the <a href="/facts/Compositum/FklJQo5j">compositum</a> <i>EF</i> is a C-extension of <i>K</i>.</li> <li>If <i>E</i> is a C-extension of <i>F</i> and <i>E</i> > <i>K</i> > <i>F</i> then <i>E</i> is a C-extension of <i>K</i>.</li></ol> <h2 id="galois-theory">Galois theory</h2> <a href="/facts/Galois_extension/52fzNQbv">Galois extension</a> A normal, separable field extension. <a href="/facts/Galois_group/A0p35p5c">Galois group</a> The <a href="/facts/Automorphism_group/65vImKwe">automorphism group</a> of a Galois extension. When it is a finite extension, this is a finite group of order equal to the degree of the extension. Galois groups for infinite extensions are <a href="/facts/Profinite_group/0RX9Da9C">profinite groups</a>. <a href="/facts/Kummer_theory/DWBtcL4A">Kummer theory</a> The Galois theory of taking <i>n</i>th roots, given enough <a href="/facts/Roots_of_unity/EqH0ho1Y">roots of unity</a>. It includes the general theory of <a href="/facts/Quadratic_extension/DWBtcL4A">quadratic extensions</a>. <a href="/facts/Artin%E2%80%93Schreier_theory/nC8hvw8k">Artin–Schreier theory</a> Covers an exceptional case of Kummer theory, in characteristic <i>p</i>. <a href="/facts/Normal_basis/3bhymK7j">Normal basis</a> A basis in the vector space sense of <i>L</i> over <i>K</i>, on which the Galois group of <i>L</i> over <i>K</i> acts transitively. <a href="/facts/Tensor_product_of_fields/FklJQo5j">Tensor product of fields</a> A different foundational piece of algebra, including the <a href="/facts/Compositum/FklJQo5j">compositum</a> operation (<a href="/facts/Join_(mathematics)/cYcLhRIR">join</a> of fields). <h2 id="extensions-of-galois-theory">Extensions of Galois theory</h2> <a href="/facts/Inverse_problem_of_Galois_theory/phfoWPLl">Inverse problem of Galois theory</a> Given a group <i>G</i>, find an extension of the rational number or other field with <i>G</i> as Galois group. <a href="/facts/Differential_Galois_theory/neKl6CLC">Differential Galois theory</a> The subject in which symmetry groups of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when <a href="/facts/Sophus_Lie/CplQVa8S">Sophus Lie</a> founded the theory of <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>. It has not, probably, reached definitive form. <a href="/facts/Grothendieck%27s_Galois_theory/NggfvWW9">Grothendieck's Galois theory</a> A very abstract approach from <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, introduced to study the analogue of the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a>. <h2 id="citations">Citations</h2> <ul><li>Adamson, Iain T. (1982). <i>Introduction to Field Theory</i> (2nd ed.). <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-28658-1.</li> <li><a href="/facts/Paul_Cohn/gnBlbFuK">Cohn, P. M.</a> (2003). <i>Basic Algebra. Groups, Rings, and Fields</i>. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-85233-587-4. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1003.00001">1003.00001</a>.</li> <li>Fried, Michael D.; Jarden, Moshe (2008). <i>Field arithmetic</i>. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-77269-9. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1145.12001">1145.12001</a>.</li> <li>Isaacs, I. Martin (1994). <i>Algebra: A Graduate Course</i>. Graduate studies in mathematics. Vol. 100. <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. p. 389. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-4799-6. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1065-7339">1065-7339</a>.</li> <li><a href="/facts/T._Y._Lam/S1ewInkE">Lam, Tsit-Yuen</a> (2005). <i>Introduction to Quadratic Forms over Fields</i>. <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>. Vol. 67. <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1095-2. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2104929">2104929</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1068.11023">1068.11023</a>.</li> <li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1997). <i>Survey of Diophantine Geometry</i>. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-61223-8. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0869.11051">0869.11051</a>.</li> <li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (2002), <i>Algebra</i>, <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, vol. 211 (Revised third ed.), New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-95385-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556">1878556</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0984.00001">0984.00001</a></li> <li>Roman, Steven (2007). <i>Field Theory</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 158. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-27678-6.</li> <li><a href="/facts/Andrzej_Schinzel/YGa5kdPN">Schinzel, Andrzej</a> (2000). <a href="https://archive.org/details/polynomialswiths0000schi"><i>Polynomials with special regard to reducibility</i></a>. Encyclopedia of Mathematics and Its Applications. Vol. 77. Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-66225-7. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0956.12001">0956.12001</a>.</li> <li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (1989). <i>Lectures on the Mordell-Weil Theorem</i>. Aspects of Mathematics. Vol. E15. Translated and edited by Martin Brown from notes by Michel Waldschmidt. Braunschweig etc.: Friedr. Vieweg & Sohn. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0676.14005">0676.14005</a>.</li> <li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (1992). <i>Topics in Galois Theory</i>. Research Notes in Mathematics. Vol. 1. Jones and Bartlett. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-86720-210-6. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0746.12001">0746.12001</a>.</li> <li>Washington, Lawrence C. (1996). <i>Introduction to Cyclotomic fields</i> (2nd ed.). New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-94762-0. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0966.11047">0966.11047</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Fried & Jarden 2008, p. 45 - Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001. <a href="https://zbmath.org/?format=complete&q=an:1145.12001" target="_blank">https://zbmath.org/?format=complete&q=an:1145.12001</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Fried & Jarden 2008, p. 214 - Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001. <a href="https://zbmath.org/?format=complete&q=an:1145.12001" target="_blank">https://zbmath.org/?format=complete&q=an:1145.12001</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Serre 1992, p. 19 - Serre, Jean-Pierre (1992). Topics in Galois Theory. Research Notes in Mathematics. Vol. 1. Jones and Bartlett. ISBN 0-86720-210-6. Zbl 0746.12001. <a href="https://zbmath.org/?format=complete&q=an:0746.12001" target="_blank">https://zbmath.org/?format=complete&q=an:0746.12001</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Schinzel 2000, p. 298 - Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications. Vol. 77. Cambridge: Cambridge University Press. ISBN 0-521-66225-7. Zbl 0956.12001. <a href="https://archive.org/details/polynomialswiths0000schi" target="_blank">https://archive.org/details/polynomialswiths0000schi</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Schinzel 2000, p. 5 - Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications. Vol. 77. Cambridge: Cambridge University Press. ISBN 0-521-66225-7. Zbl 0956.12001. <a href="https://archive.org/details/polynomialswiths0000schi" target="_blank">https://archive.org/details/polynomialswiths0000schi</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Washington 1996 - Washington, Lawrence C. (1996). Introduction to Cyclotomic fields (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0. Zbl 0966.11047. <a href="https://zbmath.org/?format=complete&q=an:0966.11047" target="_blank">https://zbmath.org/?format=complete&q=an:0966.11047</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Lam 2005, p. 342 - Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2104929" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2104929</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Fried & Jarden 2008, p. 564 - Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001. <a href="https://zbmath.org/?format=complete&q=an:1145.12001" target="_blank">https://zbmath.org/?format=complete&q=an:1145.12001</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Roman 2007, p. 46 - Roman, Steven (2007). Field Theory. Graduate Texts in Mathematics. Vol. 158. Springer-Verlag. ISBN 978-0-387-27678-6. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Lang 2002, p. 243 - Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1878556</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Fried & Jarden 2008, p. 28 - Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001. <a href="https://zbmath.org/?format=complete&q=an:1145.12001" target="_blank">https://zbmath.org/?format=complete&q=an:1145.12001</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Fried & Jarden 2008, p. 44 - Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001. <a href="https://zbmath.org/?format=complete&q=an:1145.12001" target="_blank">https://zbmath.org/?format=complete&q=an:1145.12001</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Roman 2007, p. 102 - Roman, Steven (2007). Field Theory. Graduate Texts in Mathematics. Vol. 158. Springer-Verlag. ISBN 978-0-387-27678-6. <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Isaacs 1994, p. 389 - Isaacs, I. Martin (1994). Algebra: A Graduate Course. Graduate studies in mathematics. Vol. 100. American Mathematical Society. p. 389. ISBN 0-8218-4799-6. ISSN 1065-7339. <a href="https://search.worldcat.org/issn/1065-7339" target="_blank">https://search.worldcat.org/issn/1065-7339</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Fried & Jarden 2008, p. 44 - Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001. <a href="https://zbmath.org/?format=complete&q=an:1145.12001" target="_blank">https://zbmath.org/?format=complete&q=an:1145.12001</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Roman 2007, p. 273 - Roman, Steven (2007). Field Theory. Graduate Texts in Mathematics. Vol. 158. Springer-Verlag. 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ISSN 1065-7339. <a href="https://search.worldcat.org/issn/1065-7339" target="_blank">https://search.worldcat.org/issn/1065-7339</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Lang 2002, p. 228 - Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1878556</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> </ol>