{"id":365,"date":"2012-09-18T14:25:40","date_gmt":"2012-09-18T21:25:40","guid":{"rendered":"http:\/\/zackmdavis.net\/blog\/?p=365"},"modified":"2012-09-18T14:26:43","modified_gmt":"2012-09-18T21:26:43","slug":"the-true-secret-about-conjugate-roots-and-field-automorphisms","status":"publish","type":"post","link":"http:\/\/zackmdavis.net\/blog\/2012\/09\/the-true-secret-about-conjugate-roots-and-field-automorphisms\/","title":{"rendered":"The True Secret About Conjugate Roots and Field Automorphisms"},"content":{"rendered":"<p>In the study of the elementary algebra, one occasionally hears of the conjugate roots theorem, which says that if <em>z<sub>0<\/sub><\/em> is a root of a polynomial with real coefficients, then its complex conjugate is also a root. Or if you prefer, nonreal roots come in conjugate pairs. It also works in the other direction: if nonreal roots of a polynomial come in conjugate pairs, then the polynomial has real coefficients, because the purely imaginary parts cancel when you do the algebra: (<em>x<\/em> \u2013 (<em>a<\/em> + <em>bi<\/em>))(<em>x<\/em> \u2013 (<em>a<\/em> \u2013 <em>bi<\/em>)) = <em>x<\/em><sup>2<\/sup> \u2013 <em>x<\/em>(<em>a<\/em> + <em>bi<\/em>) \u2013 <em>x<\/em>(<em>a<\/em> \u2013 <em>bi<\/em>) + (<em>a<\/em><sup>2<\/sup> \u2013 (<em>bi<\/em>)<sup>2<\/sup>) = <em>x<\/em><sup>2<\/sup> \u2013 2<em>ax<\/em> + <em>a<\/em><sup>2<\/sup> + <em>b<\/em><sup>2<\/sup>.<\/p>\n<p>There's also this idea that conjugation is the unique nontrivial &quot;well-behaved&quot; <em>automorphism<\/em> on \u2102, a map from \u2102 to itself that respects addition and multiplication: the sum (respectively product) of the conjugates is the conjugate of the sum (respectively product). The complex numbers are <em>symmetrical<\/em> around the real axis in a way that they're not around the imaginary axis: while <em>i<\/em> and \u2013<em>i<\/em> are different from <em>each other<\/em>, you can't &quot;tell which is which&quot; because they <em>behave<\/em> the same way. Contrast to 1 and \u20131, which <em>do<\/em> behave differently: if someone put either 1 or \u20131 in a box, but they wouldn't tell you which, but they <em>were<\/em> willing to tell you that &quot;The number in the box squares to itself,&quot; then you could figure out that the number in the box was 1, because \u20131 doesn't do that.<\/p>\n<p>The existence of these two ideas (the conjugate roots theorem and conjugation-as-automorphism) can't possibly be a coincidence; there must be some sense in which nonreal roots of real-coefficient polynomials come in conjugate pairs <em>because<\/em> the polynomial &quot;can't tell&quot; &quot;which is which&quot;. But it would be unsatisfying to just say this much and nothing more (&quot;<em>Theorem<\/em>: That can't possibly be a coincidence. <em>Proof<\/em> ...??&quot;); we want to say something much more general and precise. And in fact, we can\u2014<\/p>\n<p><!--more--><\/p>\n<p>Say that <em>L<\/em> is a field, and that <em>K<\/em> is a field that lives inside <em>L<\/em>, and that \u03c3 is a member of the group of field automorphisms of <em>L<\/em> that leave <em>K<\/em> alone (that is, map all members of <em>K<\/em> to themselves). Then we can show that<\/p>\n<p><em>Theorem (generalized conjugate roots theorem).<\/em> If <em>z<sub>0<\/sub><\/em> is a root of a polynomial with coefficients in <em>K<\/em>, then \u03c3(<em>z<sub>0<\/sub><\/em>) is too.<\/p>\n<p><em>Proof.<\/em> Let <span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_c8532f8caba1e4428ea19064d474f6c7.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"P(z) := \\sum_j a_jz^{j}\" \/><\/span><script type='math\/tex'>P(z) := \\sum_j a_jz^{j}<\/script> and suppose <em>P<\/em>(<em>z<sub>0<\/sub><\/em>) = 0. Then consider the value of <em>P<\/em>(\u03c3(<em>z<\/em><sub>0<\/sub>)). Precisely because \u03c3 respects multiplication, we have <span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_873aeadb33a0183f0d61666d26f214ea.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\\sum_j a_j\\sigma(z_{0})^{j} = \\sum_j a_j\\sigma(z_{0}^{j})\" \/><\/span><script type='math\/tex'>\\sum_j a_j\\sigma(z_{0})^{j} = \\sum_j a_j\\sigma(z_{0}^{j})<\/script> and because \u03c3 doesn't disturb anything in <em>K<\/em>, that's the same as <span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_8f82fd3c45125c9d509f9140e9ab2fcd.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\\sum_j \\sigma(a_j z_{0}^{j})\" \/><\/span><script type='math\/tex'>\\sum_j \\sigma(a_j z_{0}^{j})<\/script> (because <em>a<\/em>\u03c3(<em>z<\/em>) = \u03c3(<em>a<\/em>)\u03c3(<em>z<\/em>) = \u03c3(<em>az<\/em>)), and because \u03c3 respects addition, that's also the same as <span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_4a0554a36d11c5f9360c66e78a19a701.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\\sigma(\\sum_j a_j z_{0}^{j})\" \/><\/span><script type='math\/tex'>\\sigma(\\sum_j a_j z_{0}^{j})<\/script>. But <span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_832d14d21d91530bf1427eeb6271581f.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\\sigma(\\sum_j a_j z_{0}^{j}) = \\sigma(0)\" \/><\/span><script type='math\/tex'>\\sigma(\\sum_j a_j z_{0}^{j}) = \\sigma(0)<\/script>, and \u03c3(0) has to be zero for the automorphism to work. So <em>P<\/em>(\u03c3(<em>z<sub>0<\/sub><\/em>)) is zero, but that's what I've been trying to tell you this entire time. <\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the study of the elementary algebra, one occasionally hears of the conjugate roots theorem, which says that if z0 is a root of a polynomial with real coefficients, then its complex conjugate is also a root. Or if you &hellip; <a href=\"http:\/\/zackmdavis.net\/blog\/2012\/09\/the-true-secret-about-conjugate-roots-and-field-automorphisms\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[41],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/365"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=365"}],"version-history":[{"count":26,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/365\/revisions"}],"predecessor-version":[{"id":391,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/365\/revisions\/391"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=365"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=365"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=365"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}