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 <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Field functionality · AbstractAlgebra.jl</title><meta name="title" content="Field functionality · AbstractAlgebra.jl"/><meta property="og:title" content="Field functionality · AbstractAlgebra.jl"/><meta property="twitter:title" content="Field functionality · AbstractAlgebra.jl"/><meta name="description" content="Documentation for AbstractAlgebra.jl."/><meta property="og:description" content="Documentation for AbstractAlgebra.jl."/><meta property="twitter:description" content="Documentation for AbstractAlgebra.jl."/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link 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type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">AbstractAlgebra.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">AbstractAlgebra.jl</a></li><li><span class="tocitem">Fundamental interface of AbstractAlgebra.jl</span><ul><li><a class="tocitem" href="../types/">Type interface of AbstractAlgebra.jl</a></li><li><a class="tocitem" href="../visualizing_types/">Visualization of the types of AbstractAlgebra.jl</a></li><li><a class="tocitem" href="../extending_abstractalgebra/">Extending the interface of AbstractAlgebra.jl</a></li></ul></li><li><a class="tocitem" href="../constructors/">Constructing mathematical objects in AbstractAlgebra.jl</a></li><li><span class="tocitem">Rings</span><ul><li><a class="tocitem" href="../ring_introduction/">Introduction</a></li><li><a class="tocitem" href="../ring/">Ring functionality</a></li><li><a class="tocitem" href="../integer/">Integer ring</a></li><li><a class="tocitem" href="../total_fraction/">Total ring of fractions</a></li><li><a class="tocitem" href="../polynomial/">Univariate polynomial functionality</a></li><li><a class="tocitem" href="../ncpolynomial/">Univariate polynomials over a noncommutative ring</a></li><li><a class="tocitem" href="../mpolynomial/">Sparse distributed multivariate polynomials</a></li><li><a class="tocitem" href="../univpolynomial/">Universal polynomial</a></li><li><a class="tocitem" href="../laurent_polynomial/">Generic Laurent polynomials</a></li><li><a class="tocitem" href="../laurent_mpolynomial/">Sparse distributed multivariate Laurent polynomials</a></li><li><a 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href="../misc/">Miscellaneous</a></li></ul></li><li><span class="tocitem">Interfaces</span><ul><li><a class="tocitem" href="../interface_introduction/">Introduction</a></li><li><a class="tocitem" href="../ring_interface/">Ring Interface</a></li><li><a class="tocitem" href="../euclidean_interface/">Euclidean Ring Interface</a></li><li><a class="tocitem" href="../poly_interface/">Univariate Polynomial Ring Interface</a></li><li><a class="tocitem" href="../mpoly_interface/">Multivariate Polynomial Ring Interface</a></li><li><a class="tocitem" href="../series_interface/">Series Ring Interface</a></li><li><a class="tocitem" href="../residue_interface/">Residue Ring Interface</a></li><li><a class="tocitem" href="../field_interface/">Field Interface</a></li><li><a class="tocitem" href="../fraction_interface/">Fraction Field Interface</a></li><li><a class="tocitem" href="../module_interface/">Module Interface</a></li><li><a class="tocitem" href="../matrix_interface/">Matrix Interface</a></li><li><a class="tocitem" href="../map_interface/">Map Interface</a></li><li><a class="tocitem" href="../rand/">Random interface</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Fields</a></li><li class="is-active"><a href>Field functionality</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Field functionality</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/Nemocas/AbstractAlgebra.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/Nemocas/AbstractAlgebra.jl/blob/master/docs/src/field.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Field-functionality"><a class="docs-heading-anchor" href="#Field-functionality">Field functionality</a><a id="Field-functionality-1"></a><a class="docs-heading-anchor-permalink" href="#Field-functionality" title="Permalink"></a></h1><h2 id="Abstract-types-for-rings"><a class="docs-heading-anchor" href="#Abstract-types-for-rings">Abstract types for rings</a><a id="Abstract-types-for-rings-1"></a><a class="docs-heading-anchor-permalink" href="#Abstract-types-for-rings" title="Permalink"></a></h2><p>All field types in AbstractAlgebra belong to the <code>Field</code> abstract type and field elements belong to the <code>FieldElem</code> abstract type.</p><p>As Julia types cannot belong to our <code>FieldElem</code> type hierarchy, we also provide the union type <code>FieldElement</code> which includes <code>FieldElem</code> in union with the Julia types <code>Rational</code> and <code>AbstractFloat</code>.</p><p>Note that</p><pre><code class="language-julia hljs">Field &lt;: Ring
 FieldElem &lt;: RingElem
-FieldElement &lt;: RingElement</code></pre><p>Of course all <code>Ring</code> functionality is available for AbstractAlgebra fields and their elements.</p><h2 id="Functions-for-types-and-parents-of-fields"><a class="docs-heading-anchor" href="#Functions-for-types-and-parents-of-fields">Functions for types and parents of fields</a><a id="Functions-for-types-and-parents-of-fields-1"></a><a class="docs-heading-anchor-permalink" href="#Functions-for-types-and-parents-of-fields" title="Permalink"></a></h2><pre><code class="language-julia hljs">characteristic(R::MyParent)</code></pre><p>Return the characteristic of the field. If the characteristic is not known, an exception is raised.</p><h2 id="Basic-functions"><a class="docs-heading-anchor" href="#Basic-functions">Basic functions</a><a id="Basic-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Basic-functions" title="Permalink"></a></h2><pre><code class="language-julia hljs">is_unit(f::MyElem)</code></pre><p>Return <code>true</code> if the given element is invertible, i.e. nonzero in the field.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../field_introduction/">« Introduction</a><a class="docs-footer-nextpage" href="../fraction/">Generic fraction fields »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.0 on <span class="colophon-date" title="Sunday 8 October 2023 19:37">Sunday 8 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+FieldElement &lt;: RingElement</code></pre><p>Of course all <code>Ring</code> functionality is available for AbstractAlgebra fields and their elements.</p><h2 id="Functions-for-types-and-parents-of-fields"><a class="docs-heading-anchor" href="#Functions-for-types-and-parents-of-fields">Functions for types and parents of fields</a><a id="Functions-for-types-and-parents-of-fields-1"></a><a class="docs-heading-anchor-permalink" href="#Functions-for-types-and-parents-of-fields" title="Permalink"></a></h2><pre><code class="language-julia hljs">characteristic(R::MyParent)</code></pre><p>Return the characteristic of the field. If the characteristic is not known, an exception is raised.</p><h2 id="Basic-functions"><a class="docs-heading-anchor" href="#Basic-functions">Basic functions</a><a id="Basic-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Basic-functions" title="Permalink"></a></h2><pre><code class="language-julia hljs">is_unit(f::MyElem)</code></pre><p>Return <code>true</code> if the given element is invertible, i.e. nonzero in the field.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../field_introduction/">« Introduction</a><a class="docs-footer-nextpage" href="../fraction/">Generic fraction fields »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.0 on <span class="colophon-date" title="Sunday 8 October 2023 19:55">Sunday 8 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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