{"id":97,"date":"2025-09-20T01:58:47","date_gmt":"2025-09-20T01:58:47","guid":{"rendered":"https:\/\/itisallmath.com\/modern-algebra\/"},"modified":"2025-12-31T14:40:27","modified_gmt":"2025-12-31T14:40:27","slug":"modern-algebra","status":"publish","type":"page","link":"https:\/\/itisallmath.com\/es\/mathematics\/modern-algebra\/","title":{"rendered":"Modern Algebra"},"content":{"rendered":"<div class=\"wp-block-uagb-container uagb-block-a29590b5 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\"><\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-adf2686a alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1600\" height=\"423\" src=\"https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannermodernalgebra-A0xNa7q3w3hLrRMv.webp\" alt=\"\" class=\"wp-image-79\" srcset=\"https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannermodernalgebra-A0xNa7q3w3hLrRMv.webp 1600w, https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannermodernalgebra-A0xNa7q3w3hLrRMv-300x79.webp 300w, https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannermodernalgebra-A0xNa7q3w3hLrRMv-1024x271.webp 1024w, https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannermodernalgebra-A0xNa7q3w3hLrRMv-768x203.webp 768w, https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannermodernalgebra-A0xNa7q3w3hLrRMv-1536x406.webp 1536w\" sizes=\"auto, (max-width: 1600px) 100vw, 1600px\" \/><\/figure>\n\n\n\n<div class=\"wp-block-uagb-advanced-heading uagb-block-bfe058e4\"><h1 class=\"uagb-heading-text\">What are you going to learn?<\/h1><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Define a group, give examples of groups, list properties that hold in every group and state definitions of particular features of groups<\/li>\n\n\n\n<li>Define a ring, give examples of rings, list properties that hold in every ring and state definitions of particular features of rings<\/li>\n\n\n\n<li>Define a field, give examples of fields, list properties that hold in every field and state definitions of particular features of fields<\/li>\n\n\n\n<li>Prove statements about these mathematical structures<\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-uagb-advanced-heading uagb-block-34ad06fb\"><h2 class=\"uagb-heading-text\">Content<\/h2><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-07df9cb7 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-edb0fefb\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-df4567b6\"><h3 class=\"uagb-heading-text\">Chapter 1. Introduction<\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Logic, Sets, Operations<\/li>\n\n\n\n<li>Equivalence Relations<\/li>\n\n\n\n<li>Equivalence Classes<\/li>\n\n\n\n<li>Permutations Groups<\/li>\n\n\n\n<li>Subgroups<\/li>\n\n\n\n<li>Groups and Symmetry<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-b48f9561\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-8765df28 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-23ffe2df\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-a4d81650\"><h3 class=\"uagb-heading-text\"><strong>Chapter 2. Groups 1<\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Definition. Examples<\/li>\n\n\n\n<li>Permutations<\/li>\n\n\n\n<li>Subgroups<\/li>\n\n\n\n<li>Groups and Symmetry<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-a9ee8211\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-fa80aaa7 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-e010185e\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-bced2ffe\"><h3 class=\"uagb-heading-text\"><strong>Chapter 3. Groups 2<\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Properties<\/li>\n\n\n\n<li>Generators. Direct Groups<\/li>\n\n\n\n<li>Cosets<\/li>\n\n\n\n<li>Lagranges&#8217;s Theorem. Cyclic Groups<\/li>\n\n\n\n<li>Isomorphisms<\/li>\n\n\n\n<li>Cayle&#8217;s Theorem<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-e1f73288\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-5bfe47a6 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-3252887c\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-51a008f8\"><h3 class=\"uagb-heading-text\"><strong>Chapter 4. Groups Homomorphisms<\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Homomorphisms and Kernels<\/li>\n\n\n\n<li>Quotient Groups<\/li>\n\n\n\n<li>The Fundamental Homomorphism Theorem<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-aecd7001\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-52f99807 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-9e0f8538\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-3daaf221\"><h3 class=\"uagb-heading-text\"><strong>Chapter 5. Introduction to Rings<\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Definition and Examples<\/li>\n\n\n\n<li>Integral Domains, Division Rings<\/li>\n\n\n\n<li>Fields<\/li>\n\n\n\n<li>Isomorphisms. Characteristics<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-bef85f36\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-1c3b6106 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-4f66d034\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-5b46dfab\"><h3 class=\"uagb-heading-text\"><strong>Chapter 6. Numbers Systems<\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Ordered Integral Domains<\/li>\n\n\n\n<li>Integers<\/li>\n\n\n\n<li>Field of Quotients. Field of Rationals<\/li>\n\n\n\n<li>Ordered Fields. The Field of Reals<\/li>\n\n\n\n<li>The Field of Complex Numbers<\/li>\n\n\n\n<li>Complex Roots of Unity<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-e0b6ddda\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-f991879a alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-60933577\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-1081c668\"><h3 class=\"uagb-heading-text\">Chapter 7.  Polynomials<\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Definitions and Elementary properties<\/li>\n\n\n\n<li>The Division Algorithm<\/li>\n\n\n\n<li>Factorization of Polynomials<\/li>\n\n\n\n<li>Unique Factorization Domains<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-3cfb175e\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-da3c65fc alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-21ee3407\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-12f8808c\"><h3 class=\"uagb-heading-text\">Chapter 8.  Quotient Rings<\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Homomorphisms of Rings. Ideals<\/li>\n\n\n\n<li>Quotient Rings<\/li>\n\n\n\n<li>Factorization and Ideals<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-28af76b3\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-4b668fe7 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-483ec6c1\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-21022b59\"><h3 class=\"uagb-heading-text\">Chapter 9.  Galois Theory<\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Simple Extensions. Degree<\/li>\n\n\n\n<li>Roots of Polynomials<\/li>\n\n\n\n<li>Fundamental Theorem<\/li>\n\n\n\n<li>Algebraic Extensions<\/li>\n\n\n\n<li>Splitting Fields. Galois Groups<\/li>\n\n\n\n<li>Separability and Normality<\/li>\n\n\n\n<li>Fundamental Theorem of Galois Theory<\/li>\n\n\n\n<li>Solvability by Radicals<\/li>\n\n\n\n<li>Finite Fields<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-93d0001f\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-advanced-heading uagb-block-2688c1e2\"><h2 class=\"uagb-heading-text\">Bibliography<\/h2><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Durbin, J.R.<strong><em>A Modern Algebra: An Introduction<\/em>.<\/strong> 6th Ed. John Wiley and Sons, 2009.<\/li>\n\n\n\n<li>Rotman, J.J. <strong><em>A First Course in Abstract Algebra<\/em><\/strong>. 7th Ed. Prentice Hall<\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-uagb-advanced-heading uagb-block-e9361076\"><h2 class=\"uagb-heading-text\">Bibliography<\/h2><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li><\/li>\n<\/ul>\n\n\n\n<p><\/p>","protected":false},"excerpt":{"rendered":"<p>What are you going to learn? Content Chapter 1. Introduction Chapter 2. Groups 1 Chapter 3. Groups 2 Chapter 4. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":340,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"_uag_custom_page_level_css":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"disabled","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"wf_page_folders":[40],"class_list":["post-97","page","type-page","status-publish","hentry"],"_hostinger_reach_plugin_has_subscription_block":false,"_hostinger_reach_plugin_is_elementor":false,"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"hd_qu_size2":false,"1536x1536":false,"2048x2048":false,"trp-custom-language-flag":false},"uagb_author_info":{"display_name":"carroyav02@gmail.com","author_link":"https:\/\/itisallmath.com\/es\/author\/carroyav02gmail-com\/"},"uagb_comment_info":0,"uagb_excerpt":"What are you going to learn? Content Chapter 1. Introduction Chapter 2. Groups 1 Chapter 3. Groups 2 Chapter 4. [&hellip;]","_links":{"self":[{"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/pages\/97","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/comments?post=97"}],"version-history":[{"count":11,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/pages\/97\/revisions"}],"predecessor-version":[{"id":1533,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/pages\/97\/revisions\/1533"}],"up":[{"embeddable":true,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/pages\/340"}],"wp:attachment":[{"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/media?parent=97"}],"wp:term":[{"taxonomy":"wf_page_folders","embeddable":true,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/wf_page_folders?post=97"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}