{"id":109,"date":"2013-03-11T22:46:42","date_gmt":"2013-03-12T02:46:42","guid":{"rendered":"http:\/\/www.curtisbright.com\/bln\/?p=109"},"modified":"2013-03-11T22:46:42","modified_gmt":"2013-03-12T02:46:42","slug":"minimal-polynomial-misconception","status":"publish","type":"post","link":"http:\/\/localhost\/blog\/index.php\/2013\/03\/11\/minimal-polynomial-misconception\/","title":{"rendered":"Minimal polynomial misconception"},"content":{"rendered":"

Yesterday I had a discussion with a friend about computing minimal polynomials<\/a>. For example, say you are given algebraic numbers $\\alpha$ and $\\beta$ as specified by minimal polynomials which have degrees $n$ and $m$. How do you compute the minimal polynomial of $\\alpha+\\beta$, for example?
\n<\/a><\/p>\n

The method I was taught, which seems to be fairly standard (e.g., see\u00a0Dummit and Foote<\/a> Chapter 13.2) is the following:<\/p>\n

Multiply $\\alpha+\\beta$ by $\\alpha^i\\beta^j$ for each $i=0{,}~1{,}~\\dotsc{,}~n-1$ and $j=0{,}~1{,}~\\dotsc{,}~m-1$, and use the minimal polynomials of $\\alpha$ and $\\beta$ to reduce the resulting expressions to linear combinations of\u00a0$\\alpha^i\\beta^j$ (again with $0\\leq i<n$ and $0\\leq j<m$). In other words, what you are doing is computing a matrix $M$ which satisfies<\/p>\n

\\[ (\\alpha+\\beta)\\left[\\alpha^i\\beta^j\\right]_{i,j} = M\\left[\\alpha^i\\beta^j\\right]_{i,j} \\]<\/p>\n

where $[\\alpha^i\\beta^j]_{i,j}$ is a column vector containing the $\\alpha^i\\beta^j$.<\/p>\n

From this we see that $\\alpha+\\beta$ is an eigenvalue of $M$ and therefore is a root of the\u00a0characteristic\u00a0polynomial $p_M$ of $M$. If $p_M$ is irreducible then it will be the minimal polynomial of $\\alpha+\\beta$, but in general\u00a0the minimal polynomial will divide $p_M$, and so it will be necessary to factor $p_M$.<\/p>\n

However, once $p_M$ has been factored, how does one tell which factor is the required minimal polynomial? The obvious answer is to evaluate each factor at $\\alpha+\\beta$ and see which one gives zero. “You could do that numerically”, my friend says, and I respond with “…or symbolically”. But then he asks if I will always be able to determine if the symbolic expression is zero or not. Well, I hadn’t thought of that, and I admitted I wasn’t sure, but claimed “anything you can do numerically I can do symbolically!”<\/p>\n

I spent several hours yesterday trying to solve that problem, but eventually had to go to bed. This morning, after looking in Cohen<\/a> I found the following passage:<\/p>\n

…it does not make sense to ask which of the irreducible factors $\\alpha+\\beta$ is a root of, if we do not specify embeddings in $\\mathbb{C}$…<\/p><\/blockquote>\n

Wait, what? I got excited as I realized I had just uncovered a misconception of mine!<\/p>\n

Note that if the conjugates of $\\alpha$ are $\\alpha_i$ then they are all “symbolically identical”: $\\mathbb{Q}(\\alpha_i)$ is isomorphic for each conjugate. From that, I had assumed that the values of $\\alpha_i+\\beta_j$ would also be symbolically identical for all conjugates of $\\alpha$ and $\\beta$. Not true! As a trivial counterexample, if $\\alpha$ is a root of $x^3-2$ and $\\beta$ is a root of $x^3+2$ then two possible values for $\\alpha+\\beta$ are $0$ and $\\sqrt[3]{2}\\sqrt{3}i$, and these have very different algebraic behaviour.<\/p>\n

So the whole problem of computing the minimal polynomial of $\\alpha+\\beta$ was not well defined unless you specify which<\/em> roots $\\alpha$, $\\beta$ you are talking about—for example, by specifying them numerically.<\/p>\n","protected":false},"excerpt":{"rendered":"

Yesterday I had a discussion with a friend about computing minimal polynomials. For example, say you are given algebraic numbers $\\alpha$ and $\\beta$ as specified by minimal polynomials which have degrees $n$ and $m$. How do you compute the minimal … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"_links":{"self":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/109"}],"collection":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=109"}],"version-history":[{"count":0,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/109\/revisions"}],"wp:attachment":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=109"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=109"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=109"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}