Commit 45ffe9db by rrueger

### (Cleanup) Add dates to header with lecture number. Kind of messy, but works.


Signed-off-by: rrueger <rrueger@ethz.ch>
parent 593eee74
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 ... ... @@ -7,8 +7,6 @@ header-includes: | \newpage *Lecture 2019.03.08* # RECAP In **Lecture 1**, we looked at finite dimensional field extensions. ... ... @@ -28,7 +26,14 @@ We are now in posession of two "Linear Algebra" arguments: # LECTURE 3 # Fields fixed by Groups \sectionmark{\textit{2019.03.08}} \subsectionmark{\textit{Lecture 3}} \subsectionmark{\textit{Lecture 3}} ## Some Examples \subsectionmark{\textit{Lecture 3}} #### $\mathbb{C}$ over $\mathbb{R}$ Let's take the extension $\mathbb{C}$ over $\mathbb{R}$. What automorphisms of this extension do we know? Well, we know the identity and complex conjugation. Now we can ask ourselves, *can there be any more?*. Well ..., no. We know that the size of the automorphism group is bounded by the dimension of the extension: 2. ... ... @@ -192,6 +197,8 @@ These Galois extensions are very important as they have the most symmetries. --- # SUMMARY \sectionmark{\textit{2019.03.08}} \subsectionmark{\textit{Lecture 3}} **Theorem.** (Lecture 3) *Let $E$ be a field, and $H$ a subfield fixed by a group of automorphisms $\Sigma$. Then we have * \begin{align*} ... ...
 ... ... @@ -7,13 +7,18 @@ header-includes: | \newpage *Lecture 2019.03.15* # LECTURE 4 Neuen Text!! # Galois Extensions \sectionmark{\textit{2019.03.15}} \subsectionmark{\textit{Lecture 4}} In this lecture we will introduce two equivalences of the definition of a Galois extension that we saw last lecture. From now on we will refer to the inital definition as the **first equivalence**. We continue with the \subsectionmark{\textit{Lecture 4}} ## Second Equivalence \subsectionmark{\textit{Lecture 4}} **Theorem.** (Extension dimensions and Galois) *An extension is Galois if and only if* \begin{align*} ... ... @@ -31,7 +36,9 @@ The first equality is given by the the Lecture 3 Theorem. By the extension dimen This gives rise to the idea that *"Galois extensions are extensions of maximal symmetry"*. Usually the size of the automorphism group is smaller than dimension of the extension, but in the case of a Galois extension, the automorphism group as large as it can be - namely with cardinality **equal** to the dimension. \subsectionmark{\textit{Lecture 4}} ## Third Equivalence \subsectionmark{\textit{Lecture 4}} **Definition.** (Normal Field Extension) A field extension $E/F$ is **normal**, if $E$ is the splitting field over $F$ of some polynomial $p \in F[Z]$ with distinct roots. ... ... @@ -204,6 +211,8 @@ Notice how we needed to augment the field to $F(r_1, r_2)$ for the second factor Together, these ideas allow us to construct an isomorphism of $E$ that fixes $F$ and maps $r_i$ to $r_j$. This is exactly what we want. # SUMMARY \sectionmark{\textit{2019.03.15}} \subsectionmark{\textit{Lecture 4}} We now have the following equivalences of a Galois extension: ... ...
 ... ... @@ -5,13 +5,20 @@ header-includes: | \input{latex-headers.tex} --- *Lecture 2019.03.22* \newpage # LECTURE 5 # Fundamental Theorem of Galois Theory \sectionmark{\textit{2019.03.22}} \subsectionmark{\textit{Lecture 5}} We discussed three equivalent characterisations of what it means to be a Galois extension last lecture. Now we will introduce the **Fundamental Theorem of Galois Thoery**. This theorem is served in three parts. Naturally, we begin with \subsectionmark{\textit{Lecture 5}} ## Part 1 - Bijective Correspondence \subsectionmark{\textit{Lecture 5}} As always, let $E/F$ be a Galois extension, and $G = \mathrm{Gal}(E/F)$. We have seen that when there is a subextension $E/K/F$, then $\mathrm{Aut}(E/K)$ is a subgroup of $G$. We want to formalise this idea. In fact we will show that there is a bijection between the set of all subextensions of a Galois extension, and the subgroups of the Galois group. ... ... @@ -51,7 +58,9 @@ This concludes one direction of our proof. Now we proceed in the other direction This is a handy application of Galois theory, and gives us a new way to view subextensions, namely as subgroups of the automorphism group. \subsectionmark{\textit{Lecture 5}} ## Part 2 - Numerics \subsectionmark{\textit{Lecture 5}} **Thoerem.** If $E/K/F$ is a subextension of a Galois extension $E/F$, then $\mathrm{dim}(E/K) = |\mathrm{Aut}(E/K)|$. ... ... @@ -63,7 +72,9 @@ where on the left, we are referring to the group index, and on the right the ext It is a satisfying result to see that these two notations turn out to be compatible. ## Part 3 - \subsectionmark{\textit{Lecture 5}} ## Part 3 \subsectionmark{\textit{Lecture 5}} We have seen that every subextension $E/K/F$ of a Galois extension is Galois, but what about the extension $K/F$? We claim that ... ... @@ -168,6 +179,8 @@ We may conclude that $\mathrm{dim}(K/F) = |\mathrm{Aut}(K/F)|$ and thus $K/F$ is --- # SUMMARY \sectionmark{\textit{2019.03.22}} \subsectionmark{\textit{Lecture 5}} **Theorem.** Bijection ... ...
 ... ... @@ -6,13 +6,19 @@ header-includes: | --- \newpage *Lecture 2019.03.29* # LECTURE 6 - Solvability # LECTURE 6 # Solvability \sectionmark{\textit{2019.03.29}} \subsectionmark{\textit{Lecture 6}} This lecture we will investigate a very specific field extension, namely that of the rational functions in variables in $X_1, \ldots, X_n$ $\mathrm{Frac}(\mathbb{C}[X_1, \ldots, X_n])$ over the field of rational functions in the elementary symmetric functions $\mathrm{Frac}(\mathbb{C}[e_1, \ldots, e_n])$. We have seen elementary symmetric functions before in some examples, but now we will finally define them in general. \subsectionmark{\textit{Lecture 6}} ## Elementary Symmetric Functions \subsectionmark{\textit{Lecture 6}} **Definition.** (Elementary Symmetric Functions) Let $X_1, \ldots, X_n$ be variables. The set of elementary functions $\{e_0, \ldots, e_n \}$ on these $n$ variables is defined as the coefficients of the polynomial ... ... @@ -43,7 +49,9 @@ in general we have Their name stems from the fact, that owing to the nature of the product form $\prod _{k=1} ^n (Z-X_k)$, the $e_k$'s are invariant under permutation of the variables. That is $e_k (X_1, \ldots, X_n) = e_k (X_\sigma(1), \ldots, X_\sigma(n))$ for $\sigma \in S_n$. \subsectionmark{\textit{Lecture 6}} ## The extension over Elementary Symmetric Functions \subsectionmark{\textit{Lecture 6}} We can now go back to our original example, and ask ourselves is $\mathbb{C}(X_1, \ldots, X_n) / \mathbb{C}(e_1, \ldots, e_n)$ a finite extesnion? If so, is it Galois? ... ... @@ -64,7 +72,9 @@ If we now look at the action of $S_n$ on $\{X_1, \ldots, X_n \}$, with $\sigma \ \implies \mathrm{dim}(E/F) = n! \end{gather*} \subsectionmark{\textit{Lecture 6}} ## The subgroups of the extension \subsectionmark{\textit{Lecture 6}} Now that we know that$E/F$is Galois with dimension$n!$we can apply theorems from the last lecture. Specifically, to study the subextensions of$E/F\$, we can look at the subgroups of the Galois group (Part 1 of FTGT). ... ...
 ... ... @@ -3,18 +3,22 @@ header-includes: | \makeatletter \def\input@path{{source/}} \input{latex-headers.tex} \tikzset{ } --- \newpage # LECTURE 7 - Constructions with a Straightedge and Compass *2019.04.05* # LECTURE 7 # Constructions with a Straightedge and Compass \sectionmark{\textit{2019.04.05}} \subsectionmark{\textit{Lecture 7}} *Remark.* A **straightedge** refers to a ruler without markings. \subsectionmark{\textit{Lecture 7}} ## (I) Basic Operations and Constructions \subsectionmark{\textit{Lecture 7}} ### Operations ... ...
 ... ... @@ -6,7 +6,5 @@ \pagestyle{fancy} \renewcommand{\footrulewidth}{0.4pt} \fancyhead[CO,CE]{ETH Z\"urich - Algebra II - Prof. R. Pandharipande} \fancyhead[RO,RE]{} \fancyhead[LO,LE]{} \fancyfoot[CO,CE]{\href{http://git.ethz.ch/rrueger/algebra-notes}{git.ethz.ch/rrueger/algebra-notes} - \href{mailto:rrueger@ethz.ch}{rrueger@ethz.ch}} \fancyfoot[LE,RO]{\thepage}
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