What is Abstract Algebra?
<p>Abstract algebra is one of the foundational fields of modern mathematics. It contains a wide variety of subfields and has an immense number of applications. More specifically, abstract algebra is the study of <strong>algebraic structures</strong>, which are all sorts of things like <em>groups</em>, <em>rings</em>, <em>fields</em>, <em>modules</em>, <em>magmas</em>, and many more! In this article, I am going to discuss the history of abstract algebra, introduce you to some of its basic concepts, and provide some links to learn more about this fascinating area of mathematics.</p>
<h1>The History</h1>
<p>Algebra traces its roots in the study of polynomials. We can see the first evidence of polynomials dating all the way back to ancient Babylon around 1800 B.C. It is incredible how old this area of study is! There is a clay tablet from this time period called the <a href="https://en.wikipedia.org/wiki/Plimpton_322" rel="noopener ugc nofollow" target="_blank">Plimpton 322</a> which contains a list of Pythagorean triples!</p>
<p><img alt="" src="https://miro.medium.com/v2/resize:fit:700/0*lphr_MYiM6i_jau1.jpg" style="height:485px; width:700px" /></p>
<p>Plimpton 322 (<a href="https://en.wikipedia.org/wiki/Plimpton_322" rel="noopener ugc nofollow" target="_blank">Source</a>)</p>
<p>This tablet does contain a few errors, but it is remarkable how much the Babylonians were able to figure out with basic notation. Just as a reminder, a Pythagorean triple is a set of three numbers which satisfy the following equation</p>
<p>A basic example of a Pythagorean triple is (3,4,5). This concept comes from the ancient Greek mathematician <a href="https://en.wikipedia.org/wiki/Pythagoras" rel="noopener ugc nofollow" target="_blank">Pythagoras</a> who applied it to the length of sides of right triangles. It is remarkable that Babylonions knew about the equation well before then (although they likely did not apply it to triangles).</p>
<p>A significant development in algebra was the notation. Early mathematicians described equations purely in words. An equation would be denoted as “The thing plus two equals three.” This is obviously much more cumbersome than symbols, but symbolic notation that we use today was not fully used until around the 16th century and even then it took some time to catch on.</p>
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