This semester, I am teaching “Algebra II” for the first time. After “Algebra I” which covers standard “Groups, rings and fields”, this follow-up is largely Galois theory. In particular, I have to classify cyclic extensions.<\/p>\n
In the simplest case where $latex L\/K$ is a cyclic extension of degree $latex n\\geq 1$ and $latex K$ contains all $latex n$-th roots of unity (and $latex n$ is coprime to the characteristic of $latex K$), this essentially means proving that if $latex L\/K$ has cyclic Galois group of order $latex n$, then there is some $latex b\\in L$ with $latex L=K(b)$ and $latex b^n=a$ belongs to $latex K^{\\times}$. <\/p>\n
Indeed, the converse is relatively simple (in the technical sense that I can do it on paper or on the blackboard without having to think about it in advance, by just following the general principles that I remember).<\/p>\n
I had however the memory that the second step is trickier, and didn’t remember exactly how it was done. The texts I use (the notes of M. Reid<\/a>, Lang’s “Algebra” and Chambert-Loir’s delightful “Alg\u00e8bre corporelle”<\/a>, or rather its English translation) all give “the formula” for the element $latex b$ but they do not really motivate it. This is certainly rather quick, but since I can’t remember it, and yet I would like to motivate as much as possible all steps in this construction, I looked at the question a bit more carefully. <\/p>\n As it turns out, a judicious expansion and lengthening of the argument makes it (to me) more memorable and understandable. <\/p>\n The first step (which is standard and motivated by the converse) is to recognize that it is enough to find some element $latex x$ in $latex L^{\\times}$ such that $latex \\sigma(x)=\\xi x$, where $latex \\sigma$ is a generator of the Galois group $latex G=\\mathrm{Gal}(L\/K)$ and $latex \\xi $ is a primitive $latex n$-th root of unity in $latex L$. This is a statement about the $latex K$-linear action<\/i> of $latex G$ on $latex L$, or in other words about the representation of $latex G$ on the $latex K$-vector space $latex L$. So, as usual, the first question is to see what we know about this representation.<\/p>\n And we know quite a bit! Indeed, the normal basis theorem<\/i> states that $latex L$ is isomorphic to the left-regular representation of $latex G$ on the vector space $latex V$ of $latex K$-valued functions $latex \\varphi\\,:\\, G\\longrightarrow K$, which is given by The desired equation implies (because $latex G$ is generated by $latex \\sigma$) that $latex Kx$ is a sub-representation of $latex L$. In $latex V$, we have an explicit decomposition in direct sum So we just see what happens (or we remember that it works). For a character $latex \\chi\\in V$ such that $latex \\chi(\\sigma) = \\omega$, and $latex x\\in L^{\\times}$ the element corresponding to $latex \\chi$ under the $latex G$-isomorphism $latex L\\simeq V$, we obtain $latex \\sigma(x)=\\omega^{-1}x$. But by easy character theory (recall that $latex G$ is cyclic of order $latex n$) we can find $latex \\chi$ with $latex \\chi(\\sigma)=\\xi^{-1}$, and we are done.<\/p>\n I noticed that Lang hides the formula in Hilbert’s Theorem 90: an element of norm $latex 1$ in a cyclic extension, with $latex \\sigma$ a generator of the Galois group, is of the form $latex \\sigma(x)\/x$ for some non-zero $latex x$; this is applied to the $latex n$-th root of unity in $latex L$. The proof of Hilbert’s Theorem 90 uses something with the same flavor as the representation theory argument: Artin’s Lemma to the effect that the elements of $latex G$ are linearly independent as linear maps on $latex L$. I haven’t completely elucidated the parallel however.<\/p>\n (P.S. Chambert-Loir’s blog<\/a> has some recent very interesting posts on elementary Galois theory, which are highly recommended.)<\/p>\n","protected":false},"excerpt":{"rendered":" This semester, I am teaching “Algebra II” for the first time. After “Algebra I” which covers standard “Groups, rings and fields”, this follow-up is largely Galois theory. In particular, I have to classify cyclic extensions. In the simplest case where $latex L\/K$ is a cyclic extension of degree $latex n\\geq 1$ and $latex K$ contains … Continue reading Kummer extensions, Hilbert’s Theorem 90 and judicious expansion<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[157,1086,2],"tags":[],"class_list":["post-4577","post","type-post","status-publish","format-standard","hentry","category-eth","category-exercise","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/4577","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=4577"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/4577\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=4577"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=4577"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=4577"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n$latex (\\sigma\\cdot \\varphi)(\\tau)=\\varphi(\\sigma^{-1}\\tau)$.
\n(It is more usual to use the group algebra $latex K[G]$, but both are isomorphic).<\/p>\n
\n$latex V=\\bigoplus_{\\chi} K\\chi,$
\nwhere $latex \\chi$ runs over all characters $latex \\chi\\,:\\, G\\longrightarrow K$ (these really run over all characters of $latex G$ over an algebraic closure of $latex K$, because $latex K$ contains all $latex n$-th roots of unity and $latex G$ has exponent $latex n$). So $latex x$ (if it is to exist) must correspond to some character. The only thing to check now is whether we can find one with the right $latex \\sigma$ eigenvalue.<\/p>\n