Unverified Commit 45ffe9db authored by rrueger's avatar rrueger
Browse files

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


Signed-off-by: rrueger's avatarrrueger <rrueger@ethz.ch>
parent 593eee74
No preview for this file type
......@@ -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}
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment