math/abstract-algebra-charles-pinter-notes.tex
abstract-algebra-charles-pinter-notes.texRaw
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23\title{Notes on "A book of Abstract Algebra", Charles C. Pinter}
24\author{arnaucube}
25\date{February 2022}
26
27\begin{document}
28
29\maketitle
30
31\begin{abstract}
32 Notes on \emph{"A book of Abstract Algebra - by Charles C. Pinter"}, is a $LaTeX$ version of handmade notes taken while reading the book. It contains only some definitions and theorems (without proofs), so it is highly recommended to read the actual book instead of the current notes.
33 \\
34 \emph{This is an unfinished and 'work in progress' document.}
35\end{abstract}
36
37\tableofcontents
38% \newpage
39
40% TODO maybe this 'Introduction' section should be removed
41% \section{Introduction}
42%
43% $\mathbb{R} \supseteq \mathbb{Q} \supseteq \mathbb{Z} \supseteq \mathbb{N}$
44% \begin{itemize}
45% \item $\mathbb{R}$ (reals): $\{ \ldots, -3, 0, \sqrt{2}, 2, e, \pi, 4, \ldots \}$
46% \item $\mathbb{Q}$ (rationals): $\{ \frac{a}{b} \mid a, b \in \mathbb{Z} \wedge b \neq 0 \}$
47% \item $\mathbb{Z}$ (integers): $\{ \ldots, -3, -1, 0, 1, 3, \ldots \}$
48% \item $\mathbb{N}$ (naturals): non negative integers, $\{1, 2, 3, ...\}$
49% \end{itemize}
50
51\section{Groups}
52
53\begin{definition}[Group]
54 A set $G$ with an operation $*$ which satisfies the axioms:
55 \begin{enumerate}[i.]
56 \item $*$ is \emph{associative}
57 \item \emph{(identity element)} there is an element $e \in G$ s.t. $a * e = a$ and $e * a = a$ $\forall a \in G$
58 \item \emph{(inverse)} $\forall a \in G$, there is an element $a^{-1} \in G$ s.t. $a*a^{-1} = e$ and $a^{-1} * a =e$
59 \end{enumerate}
60\end{definition}
61
62\begin{definition}[Abelian Group]
63 A group $G$ is said to be \emph{commutative} if $\forall a, b \in G$, $ab = ba$. A commutative group is also called \emph{Abelian}.
64\end{definition}
65
66\begin{definition}[Order of an element]
67 In a group $G$, the order of an element $a \in G$ is the least positive integer $n$ such that $a \cdot a \cdots a = a^n = e$. It is represented by $ord(a)$.
68\end{definition}
69
70\begin{definition}[Order of a group]
71 Order of a group $G$, is the number of elements in $G$. It is represented by $|G|$.
72\end{definition}
73
74\begin{definition}[Cyclic group]
75 Let $G$ be a group, and $a \in G$. If $G$ consists of all the powers of $a$ and nothing else:
76 $$G = \{a^n : n \in \mathbb{Z}\}$$
77 then, $G$ is called a \emph{cyclic group}, and $a$ is called its \emph{generator}.
78 \\
79 The group $G$ generated by $a$ is defined by $G=\langle a \rangle$.
80\end{definition}
81
82\begin{theorem}
83 The \emph{order of a cyclic group} is the same as the \emph{order of it's generator}. In other words, for a cyclic group, $|\langle a \rangle | = ord(a)$.
84 \begin{itemize}
85 \item[] $\langle a \rangle$ defines a cyclic group generated by $a$. $\langle a \rangle = \{e, a, a^2, ..., a^{n-1}\}$
86 \item[] $| \langle a \rangle |$ defines the order of the cyclic group generated by $a$.
87 \end{itemize}
88\end{theorem}
89
90\begin{theorem}
91Every subgroup of a cyclic group is cyclic.
92\end{theorem}
93
94
95
96\section{Subgroups}
97\begin{definition}[Subgroup]
98 Let $G$ be a group, and $H$ a non-empty subset of $G$. If
99 \begin{enumerate}[i.]
100 \item the idenity $e$ of $G$ is in $H$.
101 \item $H$ is closed with respect to the operation. Which is for $a, b \in H$, $ab \in H$.
102 \item $H$ is closed with respect to inverses. Which is for $a \in H$, $a^{-1} \in H$.
103 \end{enumerate}
104 we call $H$ a \emph{subgroup} of $G$. The operation of $H$ is the same as the operation of $G$.
105\end{definition}
106
107\begin{theorem}
108 Every subgroup of a cyclic group is cyclic.
109\end{theorem}
110
111\section{Functions}
112\begin{definition}[Function]
113 If $A$ and $B$ are sets, then a function from $A$ to $B$ is a rule which to every element $x$ in $A$ assigns a unique element $y$ in $B$.
114 \\
115 Functions are represented by $f: A \rightarrow B$, where $\forall a \in A \Rightarrow f(a) \in B$.
116\end{definition}
117
118\begin{definition}[Injective (monomorphism)]
119 A function $f:A \rightarrow B$ is called \emph{injective} if each element of $B$ is the image of no more than one element of $A$.
120\end{definition}
121
122\begin{definition}[Surjective (epimorphism)]
123 A function $f:A \rightarrow B$ is called \emph{surjective} if each element of $B$ is the image of at least one element of $A$.
124\end{definition}
125
126\begin{definition}[Bijective (isomorphism)]
127 A function $f:A \rightarrow B$ is called \emph{bijective} if it is both \emph{injective} and \emph{surjective}.
128 \\
129 A function $f: A \rightarrow B$ has an inverse iff it is \emph{bijective}. In that case, the inverse $f^{-1}$ is a bijective function from $B$ to $A$.
130\end{definition}
131
132\begin{definition}[Composite function]
133 A function $f:A \rightarrow B$ and $g: B \rightarrow C$ be functions. The \emph{composite function} denoted by $g \circ f$ is a function from $A$ to $C$ defined as follows:
134 $$[g \circ f](x) = g(f(x)), \forall x \in A$$
135\end{definition}
136
137
138\begin{definition}[Permutation]
139 By a \emph{permutation} of a set $A$ we mean a \emph{bijective function from $A$ to $A$}, that is, a one-to-one correspondence between $A$ and itself.
140 \\
141 The set of all the permutations of $A$, with the operation $\circ$ of composition, is a group.
142 \\
143 For any positive integer $n$, the symmetric group on the set ${1,2, 3,..., n}$ is called the \emph{symmetric group on $n$ elements}, and is denoted by $S_n$.
144\end{definition}
145
146% TODO define 'group of permutations' & 'cycle of permutations'
147
148\section{Isomorphism}
149
150\begin{definition}[Isomorphism]
151 Let $G_1$ and $G_2$ be groups. A bijective function $f: G_1 \rightarrow G_2$ with the property that for any two elements $a, b \in G_1$,
152 $$f(ab) = f(a)f(b)$$
153 is called an \emph{isomorphism} from $G_1$ to $G_2$.
154 \\
155 If there exists an isomorphism from $G_1$ to $G_2$, we say that $G_1$ is \emph{isomorphic} to $G_2$, symbolized by $G_1 \cong G_2$.
156\end{definition}
157
158
159\begin{theorem}[Cayley's Theorem]
160 Every group is isomorphic to a group of permutations.
161\end{theorem}
162
163\begin{theorem}
164 (Isomorphism of cyclic groups)
165 \begin{enumerate}[i.]
166 \item For every positive integer $n$, every cyclic group of order $n$ is isomorphic to $\mathbb{Z}_n$. Thus, any two cyclic groups of order $n$ are isomorphic to each other.
167 \item Every cyclic group of order infinity is isomorphic to $\mathbb{Z}$, and therefore any two cyclic groups of order infinity are isomorphic to each other.
168 \end{enumerate}
169\end{theorem}
170
171\section{Cosets}
172
173\begin{definition}[Coset]
174 Let $G$ be a group, and $H$ a subgroup of $G$. For any element $a$ in $G$, the symbol $aH$ denotes the set of all products $ah$, as $a$ remains fixed and $h$ ranges over $H$. $aH$ is caled a \emph{left coset} of $H$ in $G$.
175 \\
176 In similar fashion, $Ha$ denotes the set of all products $ha$, as $a$ remains fixed an $h$ ranges over $H$. $Ha$ is called a \emph{right coset} of $H$ in $G$.
177\end{definition}
178
179\begin{theorem}
180 If $Ha$ is any coset of $H$, there is a one-to-one correspondence from $H$ to $Ha$ (there is a bijection between $H$ and $Ha$).\\
181 If $a \in G$, then $|H| = |Ha|$.
182\end{theorem}
183
184\begin{theorem}[Lagrange's theorem]
185 Let $G$ be a finite group, and $H$ any subgroup of $G$. The order of $G$ is a multiple of the order of $H$. $|H|$ divides $|G|$.
186\end{theorem}
187 Lagrange's theorem can be easily seen by the facts that:
188 \begin{enumerate}[i.]
189 \item cosets partition the group G
190 \item $|Ha| = |H|$ (each coset has the same order as H).
191 \end{enumerate}
192
193By consequence,
194\begin{theorem}
195 If $G$ is a group with a prime number $p$ of elements, then $G$ is a cyclic group. Furthermore, any element $a \neq e$ in $G$ is a generator of $G$.
196\end{theorem}
197
198Thus,
199\begin{theorem}
200 The order of any element of a finite group divides the order of the group.
201\end{theorem}
202
203\begin{definition}[Index of H in G]
204 Number of cosets of H in G. Represented by $(G:H)$.\\
205 Combined with \emph{Lagrange Theorem}, we know that $|G| = |H| \cdot |G:H|$, so,
206 $$(G:H) = \frac{|G|}{|H|}$$
207\end{definition}
208
209
210\section{Homomorphisms}
211
212\begin{definition}[Homomorhism]
213 If $G$ and $G$ are groups, a \emph{homomorphism} from $G$ to $H$ is a function $f: G \rightarrow H$ s.t. for any two elements $a, b \in G$,
214 $$f(ab) = f(a)f(b)$$
215 If there exists a homomorphism from $G$ \emph{onto} $H$, we say that $H$ is a \emph{homomorphic image} of $G$.
216\end{definition}
217Note: an \emph{isomorphism} is a \emph{bijective} \emph{homomorphism}.
218\\
219Example of an \emph{homomorphism}: $f: \mathbb{Z}_6 \rightarrow \mathbb{Z}_3$.
220
221
222\begin{theorem}
223 Let $G$ and $G$ be groups, and $f: G \rightarrow H$ a homomorphism. Then
224 \begin{enumerate}[i.]
225 \item $f(e) = e$
226 \item $f(a^{-1}) = [f(a)]^{-1}, \quad \forall a \in G$
227 \end{enumerate}
228\end{theorem}
229
230\begin{definition}[Conjugate]
231 A \emph{conjugate} of $a$ is any element of the form $xax^{-1}$, where $x \in G$.
232\end{definition}
233
234\begin{definition}[Normal subgroup]
235 Let $H$ be a subgroup of a group $G$. $H$ is called a \emph{normal} subgroup of $G$ if it is closed with respect to conjugates, that is, if\\
236 for any $a \in H$ and $x \in G$, $xax^{-1} \in H$.
237 \\
238 Alternatively, we can see that $H$ is a \emph{normal} subgroup iff $\forall a \in G, aH = Ha$.
239 \\
240 In an abelian group, every subgroup is normal.
241\end{definition}
242
243\begin{definition}[Kernel]\label{def:homomorphismkernel}
244 Let $f: G \rightarrow H$ be a homomorphism. The \emph{kernel} of $f$ is the set $K$ of all the elements of $G$ which are carried by $f$ onto the neutral element of $H$. That is,
245 $$K = {x \in G : f(x) = e}$$
246 \emph{Kernel in the context of Extension fields: \ref{def:extensionkernel}}
247\end{definition}
248For every homomorphism, the $e \in G$ maps to $e \in H$, so the \emph{kernel} is never empty, it always contains the identity $e_G$, and if the kernel only contains the identity, then $f$ is one-to-one (injective).
249
250
251\section{Quotient Groups}
252
253Quotient group construction is useful as a way of actually manufacturing all the homomorphic images of any group G. Additionally, we can often choose $H$ so as to "factor out" unwanted properties of $G$, and prserve in $G/H$ only "desirable" traits.
254
255\begin{definition}[Coset multiplication]
256 The coset of $a$, multiplied by the coset of $b$, is defined to be the coset of $ab$. In symbols, $Ha \cdot Hb = H(ab)$.
257\end{definition}
258
259\begin{theorem}
260 Let $H$ be a normal subgroup of $G$. If $Ha = Hc$ and $Hb = Hd$, then $H(ab) = H(cd)$.
261\end{theorem}
262
263\begin{definition}
264 $G/H$ denotes the set which consists of \emph{all the cosets of $H$}.
265 \\
266 Thus, if $Ha, Hb, Hc, \ldots$ are cosets of $H$, then $G/H = \{ Ha, Hb, Hc, ... \}$.
267\end{definition}
268
269\begin{theorem}[Quotient group]
270 $G/H$ with coset multiplication is a group.
271\end{theorem}
272
273\begin{theorem}
274 $G/H$ is a homomorphic image of G.
275 \\
276 Conversely, every homomorphic image of $G$ is a quotient group of $G$.
277\end{theorem}
278
279\begin{theorem}
280 Let $G$ be a group and $H$ a subgroup of $G$. Then
281 \begin{enumerate}[i.]
282 \item $Ha = Hb$ iff ${ab}^{-1} \in H$
283 \item $Ha = H$ iff $a \in H$
284 \end{enumerate}
285\end{theorem}
286
287
288\section{Rings and Fields}
289
290\begin{definition}[Ring]
291 A set $A$ with operations called \emph{addition} and \emph{multiplication} which satisfy the following axioms:
292 \begin{enumerate}[i.]
293 \item $A$ with addition alone is an abelian group.
294 \item Multiplication is associative.
295 \item Multiplication is distributive over addition. That is, $\forall a,b,c \in A$,
296 $$a(b+c) = ab + ac$$
297 $$(b+c)a = ba + ca$$
298 \end{enumerate}
299\end{definition}
300
301\begin{definition}[Commutative ring]
302 By definition, addition is commutative in every ring but multiplication is not. When multiplication also is commutative in a ring, we call that ring a \emph{commutative} ring.
303\end{definition}
304
305\begin{definition}[Unity]
306 A ring does not necessarily have a neutral element for multiplication. If there is in $A$ a neutral element for mulitplication, it is called the \emph{unity} of $A$, and is denoted by the symbol $1$.
307 \\
308 If $A$ has a unity, we call $A$ a \emph{ring with unity}.
309\end{definition}
310
311\begin{definition}[Field]
312 If $A$ is a commutative ring with unity in which every nonzero element is invertible, $A$ is called a \emph{field}.
313\end{definition}
314
315\begin{theorem}[Finite Field must be over p prime ($\mathbb{F}_p$)]
316 Proof from \href{https://github.com/aragonzkresearch/blog/blob/main/pdf/Aragon_Math_Seminar.pdf}{Matan Prasma seminars}:\\
317 One of the axioms of a field is $\exists$ multiplicative inverse.\\
318 If $\mathbb{Z}_n$ with $n$ no prime, then $n= k \cdot l$ for some $1 \leq k,~l \leq n-1$.\\
319 Then in $\mathbb{Z}_n$, $k \cdot l = 0$, but if $k \cdot l=0$ means that either $k=0$ or $l=0$ (otherwise, we could multiply by (eg) $k^{-1}$ and get $k^{-1} \cdot k \cdot l = k^{-1} \cdot 0$, which leads to $1 \cdot l = 0$).\\
320 which is a contradiction here (since $1 \leq k,~l \leq n-1$).\\
321 Thus $\mathbb{Z}_n$ with $n$ not prime can not be a field.\\
322 Conversely, if $n = p$ prime,\\
323 for $0 \neq x \in \mathbb{Z}_p$, $gcd(x, p)=1$, so Extended Euclidean Algorithm gives $u, v \in \mathbb{Z}$ such that $u x + v p = 1$.\\
324 Then, $ux=1 \pmod p$, so $u=x^{-1}$, so inverses exist.\\
325 Thus $\mathbb{Z}_p$ is a field.
326\end{theorem}
327
328\begin{definition}[Divisor of zero]
329 In any ring, a nonzero element a is called a \emph{divisor of zero} if there is a
330 nonzero element b in the ring such that the product ab or ba is equal to
331 zero.
332\end{definition}
333
334\begin{definition}[Cancellation property]
335A ring is said to have the cancellation property if $ab = ac$ or $ba = ca$ implies $b = c$
336for any elements a, b, and c in the ring if $a \neq 0$.
337\end{definition}
338
339\begin{theorem}
340 A ring has the \emph{cancellation property} iff it has no \emph{divisors of zero}.
341\end{theorem}
342
343\begin{definition}[Ideal]
344 A nonempty subset $B$ of a ring $A$ is called an \emph{ideal} of $A$ if $B$ is closed with respect to addition and negatives, and $B$ absorbs products in $A$.
345 \\
346 (\emph{Absorbs product}: $\forall b \in B$ and $x \in A$, then $xb, bx \in B$).
347\end{definition}
348
349\begin{definition}[Principal ideal]
350 A \emph{principal ideal} is an ideal $I$ in a ring $R$ that is generated by a single element $a \in R$ through multiplication by every element of $R$. In other words $I = aR = \{a r : r \in R \}$.
351 \\
352 (eg. Every ideal of $\mathbb{Z}$ is principal).
353\end{definition}
354
355\begin{definition}[Integral domain]
356 An \emph{integral domain} is defined to be a commutative ring with unity having the cancellation property.
357\end{definition}
358
359Every field is an integral domain, but the converse is not true (eg. $\mathbb{Z}$ is an integral domain but not a field).
360
361\begin{definition}[Characteristic n]
362 Let $A$ be a ring with unity, the \emph{characteristic} of $A$ is the least positive integer $n$ such that
363 $$\underbrace{1 + 1 + \cdots + 1}_{n-times} = 0$$
364 If there is no such positive integer $n$, $A$ has characteristic $0$.
365\end{definition}
366
367\section{Elements of number theory}
368
369\begin{definition}[Euclid's lemma]
370 Let $m$ and $n$ be integers, and let $p$ be a prime. If $p|(mn)$, then either $p|m$ or $p|n$.
371\end{definition}
372
373\begin{theorem}[Factorization into primes]
374 Ever integer $n>1$ can be expressed as a product of positive primes. That is, there are one or more primes $p_1, \ldots, p_r$ such that $n=p_1 p_2 \cdots p_r$.
375\end{theorem}
376
377\begin{theorem}[Unique factorization]
378 Suppose $n$ can be factored into positive primes in two ways, namely,
379 $$n= p_1 \cdots p_r = q_1 \cdots q_t$$
380 Then $r=t$, and the $p_i$ are the same numbers as the $q_j$ except, possibly, for the order in which they appear.
381\end{theorem}
382
383From the last two theorems: every integer $m$ can be factored into primes, and the prime factors of $m$ are unique (except for the order).
384
385\begin{theorem}[Little theorem of Fermat]
386 Let $p$ be a prime. Then,
387 $$a^{p-1} \equiv 1 \pmod p, \forall a \not\equiv 0 \pmod p$$
388 \\
389 So, by taking $a^{p-2} \cdot a \equiv 1 \pmod p$, where $a^{p-2} \equiv a^{-1} \pmod p$ (the inverse modulo p), we see that $a^p \equiv a \pmod p, \forall a \in \mathbb{Z}$, so $a^p - a$ is a multiple of $p$.
390
391 ~\\\emph{Relation to Lagrange's theorem:}\\
392 Let $G = \mathbb{Z}_p$, and let $H$ be the multiplicative subgroup of $G$ generated by $a$ (ie. $H = \{ 1, a, a^2, \ldots \}$). The order of $H$ ($h = |H|$), is also the order of $a$ (ie. smallest $n>1$ s.t. $a^n=1~mod~p$).
393
394 By Lagrange's theorem, $h~|~|G| = p - 1$, so $p-1 = h \cdot m$, thus
395 $$
396 a^{p-1} = (a^h)^m \equiv 1^m \equiv 1~mod~p
397 $$
398
399 ~\\\emph{Another perspective:}\\
400 We have $a^p \equiv a \pmod{p}$, by dividing by $a$ on both sides, we obtain $a^{p-1} \equiv 1 \pmod{p}$.
401
402\end{theorem}
403
404\begin{theorem}[Euler's $\phi$ function]
405 \emph{Euler's $\phi$ function} describes the number of integers in $\mathbb{Z} / n \mathbb{Z}$ which are relatively prime (coprime) to $n$.
406\end{theorem}
407
408\begin{theorem}[Euler's theorem]
409 If $a$ and $n$ are relatively prime, $$a^{\phi(n)} \equiv 1 \pmod n$$
410\end{theorem}
411
412
413\section{Polynomials}
414
415\begin{definition}
416 Let $A$ be a commutative ring with unity, and $x$ an arbitrary symbol. Every expression of the form
417 $$a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$$
418 is called a \emph{polynomial in $x$ with coefficients in $A$}, or more simply, a \emph{polynomial in $x$ over $A$}.
419\end{definition}
420 The expressions $a_k x^k$, for $k \in \{ 1, \ldots, n \}$, are called the \emph{terms} of the polynomial, being $a_n x^n$ the \emph{leading term}, and $a_0$ the \emph{constant term}.
421 The $a_k$ are called the \emph{coefficients} of $x^k$, being $a_n$ the \emph{leading coefficient}. And the \emph{degree} of a polynomial $a(x)$ is the greatest $n$ such that the coefficient of $x^n$ is not zero.
422 The polynomial whose leading coefficient is equal to $1$ is called \emph{monic}.
423
424\begin{theorem}[Division algorithm for polynomials]
425 If $a(x)$ and $b(x)$ are polynomials over a field $F$, and $b(x) \neq 0$, there exist polynomials $q(x)$ and $r(x)$ over $F$ such that
426 $a(x) = b(x) q(x) + r(x)$ and
427 [$r(x) = 0$ or $\deg r(x) < \deg b(x)$].
428\end{theorem}
429
430\begin{theorem}
431 Any two nonzero polynomials $a(x), b(x) \in F[x]$ have a $\gcd d(x)$. Furthermore, $d(x)$ can be expressed as a \emph{linear combination}
432 $$d(x)= r(x) a(x) + s(x) b(x)$$
433 where $r(x), s(x) \in F[x]$.
434\end{theorem}
435
436\begin{theorem}[Factorization into irreducible polynomials]
437 Every polynomial $a(x)$ of positive degree in $F[x]$ can be written as a product
438 $$a(x) = k p_1(x) p_2(x) \cdots p_r(x)$$
439 where $k$ is a constant in $F$ and $p_1(x), \ldots, p_r(x)$ are monic irreducible polynomials of $F[x]$.
440\end{theorem}
441
442\begin{theorem}[Unique factorization]
443 If $a(x)$ can be written in two ways as a product of monic irreducibles, say
444 $$a(x) = k p_1(x) \cdots p_r(x) = l q_1(x) \cdots q_s(x)$$
445 then $k=l$, $r=s$, and $p_i(x) = q_j(x)$.
446\end{theorem}
447
448\begin{theorem}
449 $c$ is a root of $a(x)$ iff $x - c$ is a factor of $a(x)$.
450\end{theorem}
451\begin{theorem}
452 If $a(x)$ has distinct roots $c_1, \ldots, c_m$ in $F$, then $(x-c_1)(x-c_2)\cdots(x-c_m)$ is a factor of $a(x)$.
453\end{theorem}
454
455\begin{theorem}
456 If $a(x)$ has degree $n$, it has at most $n$ roots.
457\end{theorem}
458
459In finite $F$, polynomial $\neq$ polynomial function. If $F$ is infinite, polynomial $=$ polynomial function.
460
461For every polynomial with rational coefficients, there is a polynomial with integer coefficients having the same roots. See:
462$$a(x) = \frac{k_0}{l_0} + \frac{k_1}{l_1} x + \cdots + \frac{k_n}{l_n} x^n$$
463$$=\frac{1}{l_0 \cdots l_n} \cdot \underbrace{(k_0 l_1 \cdots l_n + k_1 l_0 l_2 \cdots l_n x + \cdots + k_n l_0 \cdots l_{n-1} x^n)}_{b(x)}$$
464$a(x)$ has rational coefficients, $b(x)$ has integer coefficients. $b(x)$ differs from $a(x)$ only by a constant factor ($\frac{1}{l_0\cdots l_n}$), so $a(x)$ and $b(x)$ have the same roots.
465
466$\Longrightarrow~~\forall~p(x) \in \mathbb{Q}[x]$, there is a $f(x) \in \mathbb{R}$ with the same roots (for every polynomial with rational coefficients, there is a polynomial with integer coefficients having the same roots).
467
468\begin{theorem}
469 If $s/t$ is a root of $a(x)$, then $s|a_0$ and $t|a_n$.
470\end{theorem}
471
472\begin{theorem}
473 Suppose $a(x)$ can be factured as $a(x) = b(x)c(x)$, where $b(x), c(x)$ have rational coefficients. Then there are polynomials $B(x), C(x)$ with integer coefficients, which are constant multiples of $b(x)$ and $c(x)$ respectively, such that $a(x) = B(x)C(x)$.
474\end{theorem}
475
476\begin{theorem}[Eisenstein's irreducibility criterion]
477 Let $a(x) = a_0 + a_1 x + \cdots + a_n x^n$ be a polynomial with integer coefficients.
478
479 If there is prime $p$ such that $p | a_i, ~\forall i\in\{0, n-1\}$, and $p \ndiv a_n$ and $p^2 \ndiv a_0$, then $a(x)$ is irreducible over $\mathbb{Q}$.
480 % Suppose there is a prime number $p$ which divides every coefficient of $a(x)$ except the leading coefficient $a_n$; suppose $p$ does not divide $a_n$ and $p^2$ does not divide $a_0$. Then $a(x)$ is irreducible over $\mathbb{Q}$.
481\end{theorem}
482
483\section{Extensions of fields}
484
485\begin{definition}[Kernel]\label{def:extensionkernel}
486 The \emph{kernel} of $\sigma_c$ consists of all polynomials $a(x) \in F[x]$ such that $c$ is a root of $a(x)$.
487
488 \emph{Kernel in the context of Homomorphisms: \ref{def:homomorphismkernel}}
489\end{definition}
490
491\begin{definition}[Algebraic]
492 $c \in E$ is called \emph{algebraic over} $F$ if it is the root of some nonzero polynomial $a(x) \in F[x]$.
493
494 Otherwise, $c$ is called \emph{transcendental over} $F$.
495\end{definition}
496
497$E/K$ denotes the (field) extension of $E$ over $K$.
498
499\begin{theorem}[Basic theorem of field extensions]
500 Let $F$ be a field and $a(x) \in F[x]$ a nonconstant polynomial. There exists an extension field $E/F$ and an element $c \in E$ such that $c$ is a root of $a(x)$.
501\end{theorem}
502
503Let $a(x) \in F[x]$ be a polynomial of degree $n$. There is an extension field $E/F$ which contains all $n$ roots of $a(x)$.
504
505\section{Vector spaces}
506
507\begin{definition}[Vector space]
508 A \emph{vector space} over a field $F$ is a set $V$, with two operations $+, \cdot$, called \emph{vector addition} and \emph{scalar multiplication}, such that
509
510 \begin{itemize}
511 \item $V$ with vector addition is an abelian group
512 \item $\forall k \in F$ and $\overrightarrow{a} \in V$, the scalar product $k \overrightarrow{a}$ is an element of $V$, subject to the following conditions:
513 $\forall k, l \in F,~\overrightarrow{a},\overrightarrow{b} \in V$
514 \begin{enumerate}[i.]
515 \item $k(\overrightarrow{a} + \overrightarrow{b}) = k\overrightarrow{a} + k\overrightarrow{b}$
516 \item $(k + l)\overrightarrow{a} = k\overrightarrow{a} + k\overrightarrow{b}$
517 \item $k(l\overrightarrow{a}) = (kl)\overrightarrow{a}$
518 \item $1 \overrightarrow{a} = \overrightarrow{a}$
519 \end{enumerate}
520 \end{itemize}
521\end{definition}
522
523\begin{definition}[Linear combination]
524 If $\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n} \in V$, and $k_1, k_2, \ldots, k_n$ are scalars, then the vector
525 $$k_1 \overrightarrow{a_1} + k_2 \overrightarrow{a_2} + \cdots + k_n \overrightarrow{a_n}$$
526 is called a \emph{linear combination} of $\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}$.
527
528 The set of all the linear combinations of $\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}$ is a \emph{subspace of} $V$.
529\end{definition}
530
531\begin{definition}[Linear dependancy]
532 Let $S = \{$\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}$\}$ be a set of distinct vectors in a vector space $V$. $S$ is said to be \emph{linearly dependent} if there are scalars $k_1, \ldots, k_n$, not all zero, such that $k_1 \overrightarrow{a_1} + k_2 \overrightarrow{a_2} + \cdots + k_n \overrightarrow{a_n} = 0$.
533 Which is equivalent to saying that at least one of the vectors in $S$ is a linear combination of the others.
534
535 If $S$ is not linearly dependent, then it is \emph{linearly independent}. $S$ is linearly independent iff $k_1 \overrightarrow{a_1} + k_2 \overrightarrow{a_2} + \cdots + k_n \overrightarrow{a_n}=0$ implies $k_1 = k_2 = \cdots = k_n =0$.
536 Which is equivalent to saying thatno vector in $S$ is equal to a linear combination of the other vectors in $S$.
537\end{definition}
538
539If $\{ \overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n} \}$ is linearly dependent, then some $a_i$ is a linear combination of the preceding ones.
540
541If $\{ \overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n} \}$ spans $V$, and $a_i$ is a linear combination of the preceding vectors, then $\{ \overrightarrow{a_1}, \ldots, \crossover{\overrightarrow{a_i}}, \ldots, \overrightarrow{a_n} \}$ still spans $V$.
542
543
544\begin{theorem}
545 Any two bases of a vector space $V$ have the same number of elements.
546
547 (This comes from the fact that all bases of $\mathbb{R}^n$ contain exactly $n$ vectors)
548\end{theorem}
549
550If the set $(\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n})$ spans $V$, it contains a basis of $V$.
551
552\framebox{WIP: covered until chapter 28, work in progress.}
553
554\end{document}