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% file:         algstruct.tex
% purpose:      presentation of ``Algebraic structures'' course
% created:      pasha aug 3-9,11-12 2018
% modified:     pasha may 5 2021
% modification: fixed a typo
%

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\title{Algebraic Structures}
%\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
Version of May 5, 2021
}


\begin{document}

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\titlepage
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\begin{frame}{Literature}

\begin{itemize}

\item
S. Lang, \emph{Algebra}, revised 3rd ed., Springer, 2002
\\
(referred in what follows as {\sc Lang})

\item
S. Mac Lane and G. Birkhoff, \emph{Algebra}, 3rd ed., AMS Chelsea, 1999
\\
(referred as {\sc Mac Lane--Birkhoff})

\item
I.R. Shafarevich, Basic Notions of Algebra, Springer, 1990
\\
(referred as {\sc Shafarevich})

\end{itemize}

\vskip 0.8cm

\begin{center}
\footnotesize
(All images are courtesy of Wikipedia)
\end{center}

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\begin{frame}{}

\begin{center}
{\Huge Groups}
\end{center}

\end{frame}
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\begin{frame}{Groups}

The notion of group embodies the idea of symmetry and is one of the central 
notions in mathematics.

\begin{center}
\begin{tabular}[t]{cc}
\includegraphics[scale=0.45]{ostrava-ca.png} 
&
\includegraphics[scale=0.35]{Star_of_David.jpg} 
\\
$\mathbb Z / 5\mathbb Z$ & $\mathbb D_6$
\end{tabular}
\end{center}

\end{frame}
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\begin{frame}{Algebraic structures}

From the linear algebra course(s) you know the definition of vector space:
this is a set subject to given operations (addition, multiplication on scalars) 
satisfying given axioms (commutativity, associativity, existing of zero, etc.)

\medskip

This is an instance of an \emph{algebraic structure}: 
$\langle X, f_1, f_2, \dots \rangle$; the set $X$ with a number of (generally, 
multiary) operations $f_i: X \times \dots \times X \to X$, satisfying certain
axioms.

\medskip

Groups (and other things we will study in due course, like rings and fields) is
another instance of algebraic structures.

\end{frame}
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\begin{frame}{Groups (cont.)}

\begin{definition}
A \emph{group} is a set $G$ with a binary operation $\cdot: G \times G \to G$
(called \emph{multiplication}), and a distinguished element $e \in G$ (called the
\emph{unit}, or the \emph{neutral element}) subject to the following axioms:
\begin{itemize}
\item 
For any $a,b,c \in G$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ 
(associativity).
\item 
For any $a\in G$, $a \cdot e = e \cdot a = a$.
\item 
For any $a\in G$ there exists an element $a^{-1} \in G$ (called the 
\emph{inverse} of $a$) such that $a \cdot a^{-1} = a^{-1} \cdot a = e$.
\end{itemize}

The cardinality of the set $G$ is called an \emph{order} of the group.
\end{definition}

\end{frame}
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\begin{frame}{Examples of groups}

\begin{itemize}
\item
The \emph{trivial} group consisting of one element $e$.
\item
All symmetries of a geometric figure (i.e., continuous transformations mapping 
the figure to itself) subject to operation of composition from the first slide.
\item
All permutations of a finite set of $n$ elements, called the 
\emph{symmetric group} $S_n$.
\item
All invertible transformations of a vector space $V$ subject to operation of
composition, called the \emph{general linear group} $GL(V)$.
\item
$(\mathbb Q, +)$, $(\mathbb Q \backslash \{0\}, \times)$.

\end{itemize}

\begin{block}{Exercise 1}
What are the orders of these groups? 
\end{block}

\begin{block}{Exercise 2}
Can you give more examples? What is a general pattern for constructing them?
\end{block}

See also {\sc Shafarevich}, pp.~105,109--118,144-150.

\end{frame}
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\begin{frame}{Multiplication table}

Group can be given by enumerating elements, and specifying explicitly the table
of all products between them (called the \emph{multiplication table}).

\medskip

Groups of order $2$ and $3$:

$$\begin{array}{c|cc}
  & e & a \\
\hline
e & e & a \\
a & a & e
\end{array}$$ 

$$\begin{array}{c|ccc}
  & e & a & b \\
\hline
e & e & a & b \\
a & a & b & e \\
b & b & e & a
\end{array}$$ 

\begin{block}{Exercise}
Are there other groups of orders $2$ and $3$?
\end{block}

For a list of small groups, see {\sc Shafarevich}, p.~152.

\end{frame}
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\begin{frame}{Another exercise}

What is the minimal number $n$ such that if one removes $n$ rows and $n$ columns
from the multiplication table of an arbitrary finite group, it is always 
possible to reconstruct the group from the incomplete table?

\bigskip

{\tiny
(Proposed to, and, unfortunately, got rejected at the 2017 Vojt\v{e}ch Jarn\'ik 
International Mathematical Competition)
}

\end{frame}
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\begin{frame}{Subgroups}

Generally, a \emph{substructure} of an algebraic structure 
$\langle X, f_1, f_2, \dots \rangle$ is a subset of $X$ closed with respect to
all operations $f_1, f_2, \dots$.

\medskip

Accordingly:

\begin{definition}
A \emph{subgroup} of a group $G$ is a subset closed with respect to 
multiplication, and taking the inverse (and hence containing the unit).
\end{definition}

\medskip

(Note that distinguished elements of an algebraic structure, if any, can be 
interpreted as $0$-ary operations, thus any substructure should contain them).	

\end{frame}
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\begin{frame}{Homomorphisms and automorphisms}

Generally, a \emph{homomorphism} between two algebraic structures 
$\langle X, f_1, f_2, \dots \rangle$ and $\langle Y, f_1, f_2, \dots \rangle$
of the same signature (i.e., having the same number of operations of the same 
arity) is a map $X \to Y$ which preserves any of the operations 
$f_1, f_2, \dots$. A homomorphism is called an \emph{isomorphism} (and $X$ and
$Y$ are said \emph{isomorphic}, notationally $X \simeq Y$), if it is bijective.
An isomorphism of an algebraic structure to itself is called
\emph{automorphism}.

\bigskip

Automorphisms of any algebraic structure $X$ form a group with respect to
composition, denoted by $Aut(X)$. This is one of the reasons groups occupy a
central place among all possible algebraic structures.

\end{frame}
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\begin{frame}{Homomorphisms of groups}

According to the general notion from the previous slide:

\begin{definition}
A \emph{homomorphism} from a group $G$ to a group $H$ is a map $f: G \to H$ such
that $f(a \cdot b) = f(a) \cdot f(b)$ for any $a,b\in G$, and $f(e) = e$ (note
that, by abuse of notation, the multiplication and unit in the groups $G, H$ are
denoted by the same symbols).
\end{definition}

\begin{block}{Exercise}
Prove that if $f$ is a homomorphism of groups, then $f(a^{-1}) = f(a)^{-1}$.
\end{block}

\begin{block}{Examples of group homomorphisms}
\begin{itemize}
\item
For any $a\in G$, $x \mapsto a^{-1} x a$ is an automorphism of a group $G$, 
called an \emph{inner automorphism}.
\item
$det: GL_n(K) \to K^*$
\item
$log: (\mathbb R_{>0}, \times) \to (\mathbb R, +)$
\end{itemize}
\end{block}

\end{frame}
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\begin{frame}{Normal subgroups}

\begin{definition}
The \emph{kernel} of a group homomorphism $f: G \to H$ is the set
$\set{a\in G}{f(a) = e}$.
\end{definition}

\begin{definition}
A subgroup $N$ of a group $G$ is called \emph{normal subgroup} (denoted as
$N \triangleleft G$), if $a^{-1} x a \in N$ for any $x\in N$ and $a\in G$. 
\end{definition}

\medskip

In other words: $N$ is stable under all inner automorphisms.

\medskip

Yet in other words: any \emph{left coset} is also a \emph{right coset}:
$aN = Na$ for any $a\in G$.

\medskip

In any group $G$, the trivial subgroup, $\{e\}$, and the whole group $G$, are
normal subgroups.

\begin{definition}
All normal subgroups besides $\{e\}$ and $G$ are called \emph{proper}.
\end{definition}

\end{frame}
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\begin{frame}{Normal subgroups and quotients}

\begin{block}{Theorem}
Normal subgroups are precisely kernels of group homomorphisms.
\end{block}

(That is, the kernel of any group homomorphism is a normal subgroup, and any 
normal subgroup is the kernel of a suitable group homomorphism).

\bigskip

For a given normal subgroup $N$ of a group $G$, the homomorphism from the 
theorem is constructed as follows. Define an equivalence relation on $G$: 
$a \sim b$ if $aN = bN$. (Prove that this is an equivalence relation!).
On the set of equivalence classes -- the cosets -- define multiplication as: 
\begin{equation*}
(a N) \cdot (b N) = ab N .
\end{equation*}
(Why this is well defined?). Thus obtained structure is a group, denoted as 
$G/N$, and called the \emph{group of quotients} of $G$ by $N$.

\end{frame}
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\begin{frame}{Examples of normal subgroups and group quotients}

\begin{itemize}
\item $n\mathbb Z \triangleleft \mathbb Z$ for any integer $n$.
\item 
$GL_n(K) / \{\text{scalar matrices}\} \simeq SL_n(K)$, the group of 
$n\times n$ matrices with determinant $1$.
\end{itemize}

For more examples, see {\sc Shafarevich}, pp.~107,141 and \\ {\sc Lang}, p.~15.

\end{frame}
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\begin{frame}{Homomorphism theorems, 1 and 2}

\begin{block}{The first homomorphism theorem}
If $f: G \to H$ is a group homomorphism, then there is a group isomorphism
$G/Ker\,f \simeq Im\,f$.
\end{block}

\begin{block}{The second homomorphism theorem}
If $H$ is a normal subgroup in a group $G$, $K$ a normal subgroup both in $H$ 
and in $G$, then $H/K$ is a normal subgroup in $G/K$, and
$$
(G/K)/(H/K) \simeq G/H .
$$
\end{block}

\begin{block}{Warning}
The relation of normality in groups is not transitive! That is, if 
$K \triangleleft H \triangleleft G$, then not necessarily $K \triangleleft G$.
(Give an example!).
\end{block}

\end{frame}
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\begin{frame}{Homomorphism theorem, 3}

\begin{block}{The third homomorphism theorem}
If $S$ is a subgroup in a group $G$, and $N$ a normal subgroup in $G$, then
$N \cap S$ is a normal subgroup in $S$, $NS$ is a subgroup in $G$, and
$$
S/(N \cap S) \simeq NS/N .
$$
\end{block}

\bigskip

These three homomorphism theorems are valid in a much more general context, for
general algebraic systems.

\end{frame}
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\begin{frame}{Direct product of groups}

\begin{definition}
A \emph{direct product} of two groups $G$, $H$ is the group formed by the 
Cartesian product $G \times H$ with respect to multiplication defined as
$$
(a,b) \cdot (a^\prime, b^\prime) = (aa^\prime, bb^\prime)
$$
for any $a,a^\prime \in G$, $h,h^\prime \in H$.
\end{definition}

\begin{block}{Lemma}
$G \times \{e\}$ is a normal subgroup in $G \times H$, and
$(G \times H) / (G \times \{e\}) \simeq H$. (Similarly for $\{e\} \times H$).
\end{block}

\end{frame}
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\begin{frame}{Cyclic groups}

\begin{definition}
A group $G$ is \emph{cyclic}, if there is an element $a\in G$ such that every
element of $G$ is of the form $a^n$ for some integer $n$.
\end{definition}

\begin{block}{Theorem}
Any infinite cyclic group is isomorphic to $\mathbb Z$.
Any finite cyclic group of order $n$ is isomorphic to $\mathbb Z / n\mathbb Z$.
\end{block}

\end{frame}
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\begin{frame}{Commutator and commutant}

\begin{definition}
A \emph{commutator} of two elements $a,b$ of a group $G$ is defined as 
$a^{-1} b^{-1} a b$. A \emph{commutant} of a group $G$, denoted by $[G,G]$, is a subgroup generated
by all commutators.
\end{definition}

\end{frame}
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\begin{frame}{Abelian groups}

\begin{definition}
A group $G$ is called \emph{commutative}, or \emph{abelian}, if the 
multiplication is commutative: $a \cdot b = b \cdot a$ for any $a,b \in G$.
\end{definition}

In other words, the commutator of $G$ is trivial: $[G,G] = \{e\}$.

\medskip

``Abelian'' is in honor of Niels Henrik Abel (1802--1829):
\begin{center}
\includegraphics[scale=0.38]{Niels_Henrik_Abel.jpg}
\end{center}

\begin{block}{Exercise}
Which of the groups considered so far are abelian?
\end{block}

\end{frame}
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\begin{frame}{Nilpotent groups}

\begin{definition}
A \emph{lower central series} $G^n$ of a group $G$ is defined inductively as:
$G^1 = G$, $G^{n+1} = [G^n,G]$. A group $G$ is called \emph{nilpotent}, if
$G^n = 0$ for some $n$.
\end{definition}

\begin{block}{Example}
The group of all upper-triangular $n\times n$ matrices over a field with units 
on the diagonal.
\end{block}

\end{frame}
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\begin{frame}{Solvable groups}

\begin{definition}
A \emph{derived series} $G^{(n)}$ of a group $G$ is defined inductively as: 
$G^{(1)} = G$, $G^{(n+1)} = [G^{(n)},G^{(n)}]$. A group $G$ is called 
\emph{solvable}, if $G^{(n)} = 0$ for some $n$.
\end{definition}

The term ``solvable'' comes from Galois theory -- named after \'Evariste Galois
(1811--1832) -- which links properties of an algebraic equation with the group 
of its symmetries: an algebraic equation is solvable in radicals if and only if the group of all permutations of its
roots is solvable.
\begin{center}
\includegraphics[scale=0.11]{Evariste_galois.jpg}
\end{center}

\end{frame}
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\begin{frame}{Nilpotent and solvable groups}

\begin{block}{Lemma}
Any nilpotent group is solvable.
\end{block}

\begin{block}{Example of solvable groups}
\begin{itemize}
\item $S_3$, $S_4$.
\item The group of all upper-triangular $n\times n$ matrices over a field.
\end{itemize}
\end{block}

For more examples, see {\sc Shafarevich}, p.~156.

\begin{block}{Theorem}
A subgroup and a homomorphic image of a cyclic, respectively abelian, 
respectively nilpotent, respectively solvable group, is cyclic, respectively
abelian, respectively nilpotent, respectively solvable.

A direct product of abelian, respectively nilpotent, respectively solvable 
groups, is abelian, respectively nilpotent, respectively solvable.
\end{block}

\end{frame}
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\begin{frame}{Simple groups}

\begin{definition}
A group is called \emph{simple} if all its normal subgroups are proper.
\end{definition}

\begin{block}{Theorem}
An abelian group is simple if and only if it is isomorphic to 
$\mathbb Z / p\mathbb Z$.
\end{block}

\begin{block}{Another example}
The \emph{alternating group} $A_n$ consisting of even permutations in
$S_n$, for $n\ge 5$.
\end{block}

For more examples, see {\sc Shafarevich}, p.~157--160.

\end{frame}
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\begin{frame}{}

\begin{center}
{\Huge Rings}
\end{center}

\end{frame}
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\begin{frame}{Rings}

\begin{definition}
A \emph{ring} is a set $R$ with two binary operations 
$+, \cdot: R \times R \to R$ (called \emph{addition} and \emph{multiplication}
respectively) and distinguished element $0$ (called \emph{zero}), subject to the
following axioms:
\begin{itemize}
\item 
$R$ is an abelian group with respect to addition and with $0$ as a neutral
element.
\item 
For any $a,b,c \in R$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ 
(associativity of multiplication).
\item
For any $a\in R$, $a \cdot 0 = 0 \cdot a = 0$.
\item 
For any $a,b,c\in R$, $a\cdot (b+c) = a\cdot b + a\cdot c$ and
$(b + c)\cdot a = b\cdot a + c\cdot a$ (distributivity).
\end{itemize}
\end{definition}

\begin{block}{Exercise}
Give definitions: of a subring of a ring; of a homomorphism and isomorphism of
rings.
\end{block}

\end{frame}
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\begin{frame}{Examples of rings}

\begin{itemize}
\item Integers $\mathbb Z$.
\item 
$GF(2)$, the ring consisting of two elements $\{0,1\}$, where $1\cdot 1 = 1$
\end{itemize}

\begin{block}{Exercise}
Are there other (nonisomorphic!) rings consisting of two elements?
\end{block}

\begin{itemize}
\item 
Polynomial ring $K[x]$: all polynomials with coefficients in a field $K$.
\item
Matrix ring $M_n(K)$ of all $n\times n$ matrices with coefficients in $K$.
\item
The set of continuous real-valued functions defined on the same topological 
space.
\end{itemize}

\begin{definition}
A ring $R$ is called \emph{commutative}, if $a \cdot b = b \cdot a$ for any 
$a,b \in R$.
\end{definition}

\begin{block}{Exercise}
Give more examples. Which of all these rings are commutative?
Give examples of subrings in all these rings.
\end{block}

\end{frame}
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\begin{frame}{Examples of homomorphisms of rings}

\begin{itemize}
\item 
$\mathbb Z \to GF(2)$: even number $\mapsto 0$, odd number $\mapsto 1$.
\item
Evaluation homomorphism: from the ring of real functions to $\mathbb R$:
$f \mapsto f(x_0)$ for some fixed $x_0$ in the domain. 
\end{itemize}

\end{frame}
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\begin{frame}{Ideals}

The following goes very similarly with the group case.

\begin{definition}
The \emph{kernel} of a ring homomorphism $f: R \to S$ is the set 
$\set{a \in R}{f(a) = 0}$.
\end{definition}

\begin{definition}
An subring $I$ of a ring $R$ is called an \emph{ideal}, if $x \cdot a \in I$ and
$a \cdot x \in I$ for any $x \in I$, $a \in R$.
\end{definition}

In any ring $R$, $\{0\}$ (denoted by abuse of notation just as $0$), and the 
whole $R$ are always ideals. 

\begin{definition}
All ideals besides $0$ and $R$ are called \emph{proper ideals}.
\end{definition}

\begin{block}{Theorem}
Ideals are precisely kernels of homomorphisms.
\end{block}

\end{frame}
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\begin{frame}{Quotients}

Let $I$ be an ideal of a ring $R$. Define an equivalence relation on
$R$: $a \sim b$ if $a-b \in I$. On the set of equivalence classes define 
addition and multiplication as: 
\begin{align*}
(a+I) + (b+I) &= (a+b) + I \\
(a + I) \cdot (b + I) &= ab + I .
\end{align*}
Thus obtained structure is a ring, denoted as $R/I$, and called the 
\emph{ring of quotients} of $R$ by $I$.

\bigskip

The three homomorphism theorems similar to those for groups, are valid for 
rings.

\end{frame}
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\begin{frame}{Direct sums}

\begin{definition}
A \emph{direct sum} of two rings $R$ and $S$, denoted by $R \oplus S$, is the
ring consisting of all pairs from $R \times S$ (cartesian product) with addition
an multiplication defined component-wise:
\begin{align*}
(r,s) + (r^\prime,s^\prime) &= (r+r^\prime, s+s^\prime) \\
(r,s) \cdot (r^\prime,s^\prime) &= (r\cdot r^\prime, s\cdot s^\prime)
\end{align*}
for any $r,r^\prime \in R$, $s,s^\prime \in S$.
\end{definition}

\begin{block}{Exercise}
Establish a ring isomorphism $R \oplus S \simeq S \oplus R$.
\end{block}

\end{frame}
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\begin{frame}{}

\begin{center}
{\Huge Fields}
\end{center}

\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Fields}

\begin{definition}
A \emph{field} is a set $F$ with two binary operations 
$+, \cdot: F \times F \to F$ (called \emph{addition} and \emph{multiplication}
respectively) and two distinguished elements $0$ and $1$ (called the \emph{zero}
and the \emph{unit} respectively), subject to the following axioms:
\begin{itemize}
\item $F$ is a commutative ring with respect to addition and multiplication.
\item For any $a\in F$, $a \cdot 1 = a$.
\item 
For any $a\in F$, there is an element $a^{-1} \in F$ (called the \emph{inverse}
of $a$), such that $a \cdot a^{-1} = 1$.
\end{itemize}
\end{definition}

\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Fields (cont.)}

In other words, $F$ is an abelian group with respect to addition (called the
\emph{additive group} of $F$), and $F^* = F \backslash \{0\}$ is an abelian 
group with respect to multiplication (called the \emph{multiplicative group} of
$F$).

\bigskip

Yet in other words, $F$ is a commutative ring with unit all whose
nonzero elements are invertible.

\bigskip

\begin{block}{Exercise}
Give definitions: of a subfield of a field; of an isomorphism of fields.
Why we do not speak about homomorphism of fields?
\end{block}

\begin{block}{Theorem}
A commutative ring with unit is a field if and only if it does not have any 
proper ideals.
\end{block}

\end{frame}
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\begin{frame}{Examples}

\begin{block}{Examples of fields}

\begin{itemize}
\item
Number fields: rational numbers $\mathbb Q$, real numbers $\mathbb R$, 
complex numbers $\mathbb C$.
We have a chain of subfields: $\mathbb Q \subset \mathbb R \subset \mathbb C$.

\item
$GF(2)$ is actually a field, the smallest possible one.

\item
The polynomial ring $K[x]$ is not a field. (Prove this!). But if we consider 
$K(x)$, the set of all formal ``fractions'' with elements from $K[x]$, i.e., all
expressions of the form $\frac{f(x)}{g(x)}$, where $f(x), g(x) \in K[x]$, with 
the usual rules for addition and multiplication of fractions, then it becomes a
field. (Prove this!)
\end{itemize}

\end{block}

\begin{block}{Example of a field isomorphism}
$\mathbb R[x] / (x^2+1) \simeq \mathbb C$
\end{block}

\end{frame}
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\begin{frame}{Algebraically closed fields}

\begin{definition}
A field $F$ is called \emph{algebraically closed}, if every polynomial with 
coefficients in $F$ has a root in $F$
\end{definition}

\begin{block}{Examples}
$\mathbb C$ is algebraically closed, while $\mathbb R$ or finite fields are 
not.
\end{block}

\begin{block}{Theorem}
Any field has an algebraically closed extension.
\end{block}

\begin{definition}
Given a field $F$, such minimal extension is called an \emph{algebraic closure}
of $F$, and is denoted by $\overline F$. 
\end{definition}

\begin{block}{Example}
$\overline{\mathbb R} = \mathbb C$
\end{block}

\end{frame}
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\begin{frame}{Characteristic}

\begin{definition}
A \emph{characteristic} of a field is the minimal number $n$ such that
$\underbrace{1 + \dots + 1}_{n \text{ times}} = 0$, or zero if such number does
not exist. 
\end{definition}

\begin{block}{Theorem}
A characteristic of a field is either $0$, or a prime number.
\end{block}

\begin{block}{Examples}
Characteristic of $\mathbb Q$, $\mathbb R$, and $\mathbb C$ is zero, 
characteristic of $GF(2)$ is $2$.
\end{block}

\end{frame}
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\begin{frame}{Finite fields}

We have already encountered a finite field of $2$ elements, $GF(2)$.

\begin{block}{Theorem 1}
The ring quotient $\mathbb Z / n\mathbb Z$ is a field if and only if $n$ is 
prime.
\end{block}

\medskip

This field is denoted by $GF(p)$. 

\begin{block}{Theorem 2}
\begin{enumerate}
\item 
The number of elements of a finite field has the form $p^n$, where $p$
is the characteristic of the field;
\item 
For each prime $p$ and positive integer $n$ there exists a field of $p^n$ 
elements;
\item
Two finite fields with the same number of elements are isomorphic.
\end{enumerate}
\end{block}

The unique, according to this theorem, field of $p^n$ elements is denoted by
$GF(p^n)$. (``GF'' stands for ``Galois field'').

\begin{block}{Exercise}
Draw multiplication tables for the field $GF(4)$.
\end{block}

\end{frame}
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\begin{frame}{Prime subfields}

\begin{block}{Theorem}
Every field $F$ contains a minimal subfield $P$ (i.e., any other subfield of
$F$ contains $P$), which is isomorphic either to $\mathbb Q$ (if characteristic
of $F$ is zero), or to $GF(p)$ (of characteristic of $F$ is $p$).
\end{block}

\begin{definition}
This subfield $P$ is called a \emph{prime subfield} of $F$.
\end{definition}

\end{frame}
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\begin{frame}{}

\begin{center}
{\Huge Algebras}
\end{center}

\end{frame}
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\begin{frame}{Algebras}

Algebra = ring + vector space.

\begin{definition}
An \emph{algebra} over a field $K$ is a vector space $A$ over a field $K$ with
a binary operation $\cdot: A \times A \to A$ such that $(A,+,\cdot)$ is a ring,
and additionally, the following axiom is satisfied: 
$\lambda (a \cdot b) = (\lambda a) \cdot b$ for any $a,b\in A$, $\lambda \in K$.
\end{definition}

\begin{block}{Terminological warning}
``Algebra'' has (at least) two (related) meanings: first, a part of mathematics,
and second, a particular mathematical structure (which is studied in the scope 
of algebra in the first sense).
\end{block}

\end{frame}
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\begin{frame}{Example of algebras}

\begin{itemize}
\item
The polynomial ring $K[x]$, and the matrix ring $M_n(K)$ are, actually, algebras
over the field $K$.
\item
Power series $K[[x]]$ over a field $K$.
\end{itemize}

\end{frame}
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\begin{frame}{Structure constants}

If we fix a basis $\{e_i\}_{i\in I}$ in an algebra $A$, then the multiplication
in $A$ is uniquely determined by multiplication between all pairs of the 
elements of the basis (``multiplication table''):
$$
e_i \cdot e_j = \sum_{i\in I} C_{ij}^k e_k ,
$$
where $i,j\in I$. The elements of the base field $\{C_{ij}^k\}_{i,j,k\in I}$ 
are called \emph{structure constants} of $A$ in the basis $\{e_i\}$.

\begin{block}{Example}
The matrix units $E_{ij}$, $i,j=1,\dots,n$ form a basis of the matrix algebra
$M_n(K)$. The multiplication table in this basis is
$$
E_{ij} E_{k\ell} = \delta_{jk} E_{i\ell} .
$$
\end{block}

\begin{block}{Exercise}
Take a few low-dimensional algebras and write their structure constants in some
bases.
\end{block}


\end{frame}
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\begin{frame}{Quaternions}

\begin{block}{Definition}
The \emph{quaternion algebra} $\mathbb H$ over a field $K$ is the 
$4$-dimensional algebra with the basis $\{1,i,j,k\}$, where $1$ is the unit, and
the rest of multiplication table is \\
$i^2 = j^2 = k^2 = -1, ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j$
\vskip -5pt
\end{block}
\begin{center}
\includegraphics[scale=0.80]{quat-bridge.jpg}
\end{center}
\vskip -5pt
For more about quaternions and why they are important, see 
{\sc Shafarevich}, pp.~65--66 and {\sc Mac Lane--Birkhoff}, pp.~281--283.

\end{frame}
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\begin{frame}{}

\begin{center}
{\Huge Modules}
\end{center}

\end{frame}
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\begin{frame}{Modules}

Modules is a generalization of the concept of vector space, where instead of
the underlying field we are taking a ring: the axioms are exactly the same as
for the vector space, with multiplication by elements of a field being replaced
by multiplication by element of a ring.

\begin{block}{Examples}
\begin{itemize}
\item A ring is a module over itself.
\item More generally, an ideal in a ring is module over that ring.
\item 
The vector space of vector fields (say, in the $2$-dimensional real space)
$\{f(x,y)\partial / \partial x + g(x,y)\partial / \partial y\}$ 
is a module over a ring of functions.
\item
Abelian groups are nothing but modules over $\mathbb Z$.
\end{itemize}
\end{block}

For more examples see {\sc Shafarevich}, pp.~34--36.

\end{frame}
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\begin{frame}{Modules (cont.)}

\begin{block}{Exercise}
Give definitions of a submodule, a homomorphism of modules, and a quotient 
module.
\end{block}

\end{frame}
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\begin{frame}{Tensor product}

The tensor product is an analog of ``multiplication'' for modules.

\begin{block}{Theorem}
Let $M$, $N$ be two modules over a ring $R$. Consider a class of modules $L$ 
such that there is a ``multiplication'' map 
$M \times N \to L$, $(x,y) \mapsto xy$ satisfying the following bilinearity 
properties:
\begin{gather*}
(x_1 + x_2)y = x_1 y + x_2 y \text{ for any } x_1,x_2 \in M, y \in N \\
x(y_1 + y_2) = x y_1 + x y_2 \text{ for any } x \in M, y_1,y_2 \in M \\
(ax)y = x(ay) = a(xy) \text{ for any } a \in R, x \in M, y \in N .
\end{gather*}
Then there is a ``universal'' module in this class, denoted by $M\otimes_R N$
with multiplication denoted by $x \otimes y$: for any other module $L$ in the 
class, there exists a unique homomorphism $\varphi: M\otimes_R N \to L$ such 
that $xy = \varphi(x \otimes y)$.
\end{block}

\begin{definition}
The module $M \otimes_R N$ is called the \emph{tensor product} of the modules 
$M$ and $N$.
\end{definition}

\end{frame}
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\begin{frame}{Tensor product (cont.)}

The tensor product $M \otimes_R N$ can be constructed explicitly as a submodule
of the $R$-module generated by $M \times N$ subject to the following relations:
\begin{gather*}
(x_1 + x_2,y) - (x_1,y) - (x_2,y) \\
(x,y_1 + y_2) - (x,y_1) - (x,y_2) \\
a(x,y) - (x,ay) \\
a(x,y) - (ax,y) .
\end{gather*}

\begin{block}{Theorem}
If $R$ is a field (i.e., $M$ and $N$ are vector spaces over $R$, and 
$\{e_i\}_{i\in I}$ and $\{f_i\}_{i\in J}$ are $R$-bases of $M$ and $N$ 
respectively, then $\{e_i \otimes f_j\}_{i\in I, j\in J}$ is a basis of the
$R$-vector space $M\otimes_R N$.
\end{block}

\begin{block}{Corollary}
If $R$ is a field, then $\dim (M \otimes_R N) = \dim M \cdot \dim N$.
\end{block} 

\end{frame}
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\begin{frame}{Tensor product (cont.)}

\begin{block}{Warning}
The intuition for the tensor product of vector spaces works badly in the general
case. For example, if $n$ and $m$ are relatively prime, then 
$\mathbb Z / n\mathbb Z \otimes_{\mathbb Z} \mathbb Z / m\mathbb Z = 0$. (Prove
this!)
\end{block}

\bigskip

We can go further and iterate the tensor product construction. In particular,
tensoring the same module $M$ with itself, and taking direct sums of all tensor
powers, we get the \emph{tensor algebra} of $M$, and taking in it various 
quotients and subspaces we get symmetric and skew-symmetric powers, etc.
See {\sc Shafarevich}, pp.~38--39, {\sc Lang}, pp.~601--612, 632--637, and
{\sc Mac Lane--Birkhoff}, pp.~522--547 for details.

\end{frame}
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\begin{frame}

\begin{center}
{\Huge The End}
\end{center}

\end{frame}
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\end{document}
% eof
