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 \\ In other words, does not repeat outputs.
125\end{definition}
126
127\begin{definition}[Bijective (isomorphism)]
128 A function $f:A \rightarrow B$ is called \emph{bijective} if it is both \emph{injective} and \emph{surjective}.
129 \\
130 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 $A$.
131\end{definition}
132
133In finite sets, if $f: A \rightarrow B$ is injective then $|A| \leq |B|$, and if $f$ is surjective then $|B| \leq |A|$. And if $f$ is bijective, then $|A| = |B|$.
134
135\begin{definition}[Composite function]
136 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:
137 $$[g \circ f](x) = g(f(x)), \forall x \in A$$
138\end{definition}
139
140
141\begin{definition}[Permutation]
142 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.
143 \\
144 The set of all the permutations of $A$, with the operation $\circ$ of composition, is a group.
145 \\
146 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 $S_n$.
147\end{definition}
148
149% TODO define 'group of permutations' & 'cycle of permutations'
150
151\section{Isomorphism}
152
153\begin{definition}[Isomorphism]
154 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 $a, b \in G_1$,
155 $$f(ab) = f(a)f(b)$$
156 is called an \emph{isomorphism} from $G_1$ to $G_2$.
157 \\
158 If there exists an isomorphism from $G_1$ to $G_2$, we say that $G_1$ is \emph{isomorphic} to $G_2$, symbolized by $G_1 \cong G_2$.
159\end{definition}
160
161
162\begin{theorem}[Cayley's Theorem]
163 Every group is isomorphic to a group of permutations.
164\end{theorem}
165
166\begin{theorem}
167 (Isomorphism of cyclic groups)
168 \begin{enumerate}[i.]
169 \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.
170 \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.
171 \end{enumerate}
172\end{theorem}
173
174\section{Cosets}
175
176\begin{definition}[Coset]
177 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 $G$.
178 \\
179 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 $G$.
180\end{definition}
181
182\begin{theorem}
183 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 $Ha$).\\
184 If $a \in G$, then $|H| = |Ha|$.
185\end{theorem}
186
187\begin{theorem}[Lagrange's theorem]
188 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 $|G|$.
189\end{theorem}
190 Lagrange's theorem can be easily seen by the facts that:
191 \begin{enumerate}[i.]
192 \item cosets partition the group G
193 \item $|Ha| = |H|$ (each coset has the same order as H).
194 \end{enumerate}
195
196By consequence,
197\begin{theorem}
198 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 $G$.
199\end{theorem}
200
201Thus,
202\begin{theorem}
203 The order of any element of a finite group divides the order of the group.
204\end{theorem}
205
206\begin{definition}[Index of H in G]
207 Number of cosets of H in G. Represented by $(G:H)$.\\
208 Combined with \emph{Lagrange Theorem}, we know that $|G| = |H| \cdot |G:H|$, so,
209 $$(G:H) = \frac{|G|}{|H|}$$
210\end{definition}
211
212
213\section{Homomorphisms}
214
215\begin{definition}[Homomorhism]
216 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 $a, b \in G$,
217 $$f(ab) = f(a)f(b)$$
218 If there exists a homomorphism from $G$ \emph{onto} $H$, we say that $H$ is a \emph{homomorphic image} of $G$.
219\end{definition}
220Note: an \emph{isomorphism} is a \emph{bijective} \emph{homomorphism}.
221\\
222Example of an \emph{homomorphism}: $f: \mathbb{Z}_6 \rightarrow \mathbb{Z}_3$.
223
224
225\begin{theorem}
226 Let $G$ and $G$ be groups, and $f: G \rightarrow H$ a homomorphism. Then
227 \begin{enumerate}[i.]
228 \item $f(e) = e$
229 \item $f(a^{-1}) = [f(a)]^{-1}, \quad \forall a \in G$
230 \end{enumerate}
231\end{theorem}
232
233\begin{definition}[Conjugate]
234 A \emph{conjugate} of $a$ is any element of the form $xax^{-1}$, where $x \in G$.
235\end{definition}
236
237\begin{definition}[Normal subgroup]
238 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\\
239 for any $a \in H$ and $x \in G$, $xax^{-1} \in H$.
240 \\
241 Alternatively, we can see that $H$ is a \emph{normal} subgroup iff $\forall a \in G, aH = Ha$.
242 \\
243 In an abelian group, every subgroup is normal.
244\end{definition}
245
246\begin{definition}[Kernel]\label{def:homomorphismkernel}
247 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,
248 $$K = {x \in G : f(x) = e}$$
249 \emph{Kernel in the context of Extension fields: \ref{def:extensionkernel}}
250\end{definition}
251For 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).
252
253
254\section{Quotient Groups}
255
256Quotient 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.
257
258\begin{definition}[Coset multiplication]
259 The coset of $a$, multiplied by the coset of $b$, is defined to be the coset of $ab$. In symbols, $Ha \cdot Hb = H(ab)$.
260\end{definition}
261
262\begin{theorem}
263 Let $H$ be a normal subgroup of $G$. If $Ha = Hc$ and $Hb = Hd$, then $H(ab) = H(cd)$.
264\end{theorem}
265
266\begin{definition}
267 $G/H$ denotes the set which consists of \emph{all the cosets of $H$}.
268 \\
269 Thus, if $Ha, Hb, Hc, \ldots$ are cosets of $H$, then $G/H = \{ Ha, Hb, Hc, ... \}$.
270\end{definition}
271
272\begin{theorem}[Quotient group]
273 $G/H$ with coset multiplication is a group.
274\end{theorem}
275
276\begin{theorem}
277 $G/H$ is a homomorphic image of G.
278 \\
279 Conversely, every homomorphic image of $G$ is a quotient group of $G$.
280\end{theorem}
281
282\begin{theorem}
283 Let $G$ be a group and $H$ a subgroup of $G$. Then
284 \begin{enumerate}[i.]
285 \item $Ha = Hb$ iff ${ab}^{-1} \in H$
286 \item $Ha = H$ iff $a \in H$
287 \end{enumerate}
288\end{theorem}
289
290
291\section{Rings and Fields}
292
293\begin{definition}[Ring]
294 A set $A$ with operations called \emph{addition} and \emph{multiplication} which satisfy the following axioms:
295 \begin{enumerate}[i.]
296 \item $A$ with addition alone is an abelian group.
297 \item Multiplication is associative.
298 \item Multiplication is distributive over addition. That is, $\forall a,b,c \in A$,
299 $$a(b+c) = ab + ac$$
300 $$(b+c)a = ba + ca$$
301 \end{enumerate}
302\end{definition}
303
304\begin{definition}[Commutative ring]
305 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.
306\end{definition}
307
308\begin{definition}[Unity]
309 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 $1$.
310 \\
311 If $A$ has a unity, we call $A$ a \emph{ring with unity}.
312\end{definition}
313
314\begin{definition}[Field]
315 If $A$ is a commutative ring with unity in which every nonzero element is invertible, $A$ is called a \emph{field}.
316\end{definition}
317
318\begin{theorem}[Finite Field must be over p prime ($\mathbb{F}_p$)]
319 Proof from \href{https://github.com/aragonzkresearch/blog/blob/main/pdf/Aragon_Math_Seminar.pdf}{Matan Prasma seminars}:\\
320 One of the axioms of a field is $\exists$ multiplicative inverse.\\
321 If $\mathbb{Z}_n$ with $n$ no prime, then $n= k \cdot l$ for some $1 \leq k,~l \leq n-1$.\\
322 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 $1 \cdot l = 0$).\\
323 which is a contradiction here (since $1 \leq k,~l \leq n-1$).\\
324 Thus $\mathbb{Z}_n$ with $n$ not prime can not be a field.\\
325 Conversely, if $n = p$ prime,\\
326 for $0 \neq x \in \mathbb{Z}_p$, $gcd(x, p)=1$, so Extended Euclidean Algorithm gives $u, v \in \mathbb{Z}$ such that $u x + v p = 1$.\\
327 Then, $ux=1 \pmod p$, so $u=x^{-1}$, so inverses exist.\\
328 Thus $\mathbb{Z}_p$ is a field.
329\end{theorem}
330
331\begin{definition}[Divisor of zero]
332 In any ring, a nonzero element a is called a \emph{divisor of zero} if there is a
333 nonzero element b in the ring such that the product ab or ba is equal to
334 zero.
335\end{definition}
336
337\begin{definition}[Cancellation property]
338A ring is said to have the cancellation property if $ab = ac$ or $ba = ca$ implies $b = c$
339for any elements a, b, and c in the ring if $a \neq 0$.
340\end{definition}
341
342\begin{theorem}
343 A ring has the \emph{cancellation property} iff it has no \emph{divisors of zero}.
344\end{theorem}
345
346\begin{definition}[Ideal]
347 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 $A$.
348 \\
349 (\emph{Absorbs product}: $\forall b \in B$ and $x \in A$, then $xb, bx \in B$).
350\end{definition}
351
352\begin{definition}[Principal ideal]
353 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 $I = aR = \{a r : r \in R \}$.
354 \\
355 (eg. Every ideal of $\mathbb{Z}$ is principal).
356\end{definition}
357
358\begin{definition}[Integral domain]
359 An \emph{integral domain} is defined to be a commutative ring with unity having the cancellation property.
360\end{definition}
361
362Every field is an integral domain, but the converse is not true (eg. $\mathbb{Z}$ is an integral domain but not a field).
363
364\begin{definition}[Characteristic n]
365 Let $A$ be a ring with unity, the \emph{characteristic} of $A$ is the least positive integer $n$ such that
366 $$\underbrace{1 + 1 + \cdots + 1}_{n-times} = 0$$
367 If there is no such positive integer $n$, $A$ has characteristic $0$.
368\end{definition}
369
370\section{Elements of number theory}
371
372\begin{definition}[Euclid's lemma]
373 Let $m$ and $n$ be integers, and let $p$ be a prime. If $p|(mn)$, then either $p|m$ or $p|n$.
374\end{definition}
375
376\begin{theorem}[Factorization into primes]
377 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 $n=p_1 p_2 \cdots p_r$.
378\end{theorem}
379
380\begin{theorem}[Unique factorization]
381 Suppose $n$ can be factored into positive primes in two ways, namely,
382 $$n= p_1 \cdots p_r = q_1 \cdots q_t$$
383 Then $r=t$, and the $p_i$ are the same numbers as the $q_j$ except, possibly, for the order in which they appear.
384\end{theorem}
385
386From the last two theorems: every integer $m$ can be factored into primes, and the prime factors of $m$ are unique (except for the order).
387
388\begin{theorem}[Little theorem of Fermat]
389 Let $p$ be a prime. Then,
390 $$a^{p-1} \equiv 1 \pmod p, \forall a \not\equiv 0 \pmod p$$
391 \\
392 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 $p$.
393
394 ~\\\emph{Relation to Lagrange's theorem:}\\
395 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. $a^n=1~mod~p$).
396
397 By Lagrange's theorem, $h~|~|G| = p - 1$, so $p-1 = h \cdot m$, thus
398 $$
399 a^{p-1} = (a^h)^m \equiv 1^m \equiv 1~mod~p
400 $$
401
402 ~\\\emph{Another perspective:}\\
403 We have $a^p \equiv a \pmod{p}$, by dividing by $a$ on both sides, we obtain $a^{p-1} \equiv 1 \pmod{p}$.
404
405\end{theorem}
406
407\begin{theorem}[Euler's $\phi$ function]
408 \emph{Euler's $\phi$ function} describes the number of integers in $\mathbb{Z} / n \mathbb{Z}$ which are relatively prime (coprime) to $n$.
409\end{theorem}
410
411\begin{theorem}[Euler's theorem]
412 If $a$ and $n$ are relatively prime, $$a^{\phi(n)} \equiv 1 \pmod n$$
413\end{theorem}
414
415
416\section{Polynomials}
417
418\begin{definition}
419 Let $A$ be a commutative ring with unity, and $x$ an arbitrary symbol. Every expression of the form
420 $$a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$$
421 is called a \emph{polynomial in $x$ with coefficients in $A$}, or more simply, a \emph{polynomial in $x$ over $A$}.
422\end{definition}
423 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}.
424 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.
425 The polynomial whose leading coefficient is equal to $1$ is called \emph{monic}.
426
427\begin{theorem}[Division algorithm for polynomials]
428 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
429 $a(x) = b(x) q(x) + r(x)$ and
430 [$r(x) = 0$ or $\deg r(x) < \deg b(x)$].
431\end{theorem}
432
433\begin{theorem}
434 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}
435 $$d(x)= r(x) a(x) + s(x) b(x)$$
436 where $r(x), s(x) \in F[x]$.
437\end{theorem}
438
439\begin{theorem}[Factorization into irreducible polynomials]
440 Every polynomial $a(x)$ of positive degree in $F[x]$ can be written as a product
441 $$a(x) = k p_1(x) p_2(x) \cdots p_r(x)$$
442 where $k$ is a constant in $F$ and $p_1(x), \ldots, p_r(x)$ are monic irreducible polynomials of $F[x]$.
443\end{theorem}
444
445\begin{theorem}[Unique factorization]
446 If $a(x)$ can be written in two ways as a product of monic irreducibles, say
447 $$a(x) = k p_1(x) \cdots p_r(x) = l q_1(x) \cdots q_s(x)$$
448 then $k=l$, $r=s$, and $p_i(x) = q_j(x)$.
449\end{theorem}
450
451\begin{theorem}
452 $c$ is a root of $a(x)$ iff $x - c$ is a factor of $a(x)$.
453\end{theorem}
454\begin{theorem}
455 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 $a(x)$.
456\end{theorem}
457
458\begin{theorem}
459 If $a(x)$ has degree $n$, it has at most $n$ roots.
460\end{theorem}
461
462In finite $F$, polynomial $\neq$ polynomial function. If $F$ is infinite, polynomial $=$ polynomial function.
463
464For every polynomial with rational coefficients, there is a polynomial with integer coefficients having the same roots. See:
465$$a(x) = \frac{k_0}{l_0} + \frac{k_1}{l_1} x + \cdots + \frac{k_n}{l_n} x^n$$
466$$=\frac{1}{l_0 \cdots l_n} \cdot \underbrace{(k_0 l_1 \cdots l_n + k_1 l_0 l_2 \cdots l_n x + \cdots + k_n l_0 \cdots l_{n-1} x^n)}_{b(x)}$$
467$a(x)$ 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.
468
469$\Longrightarrow~~\forall~p(x) \in \mathbb{Q}[x]$, 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).
470
471\begin{theorem}
472 If $s/t$ is a root of $a(x)$, then $s|a_0$ and $t|a_n$.
473\end{theorem}
474
475\begin{theorem}
476 Suppose $a(x)$ can be factored 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 $a(x) = B(x)C(x)$.
477\end{theorem}
478
479\begin{theorem}[Eisenstein's irreducibility criterion]
480 Let $a(x) = a_0 + a_1 x + \cdots + a_n x^n$ be a polynomial with integer coefficients.
481
482 If there is a 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 $\mathbb{Q}$.
483 % 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 $\mathbb{Q}$.
484\end{theorem}
485
486\section{Extensions of fields}
487
488\begin{definition}[Kernel]\label{def:extensionkernel}
489 The \emph{kernel} of $\sigma_c$ consists of all polynomials $a(x) \in F[x]$ such that $c$ is a root of $a(x)$.
490
491 \emph{Kernel in the context of Homomorphisms: \ref{def:homomorphismkernel}}
492\end{definition}
493
494\begin{definition}[Algebraic]
495 $c \in E$ is called \emph{algebraic over} $F$ if it is the root of some nonzero polynomial $a(x) \in F[x]$.
496
497 Otherwise, $c$ is called \emph{transcendental over} $F$.
498\end{definition}
499
500$E/K$ denotes the (field) extension of $E$ over $K$.
501
502\begin{theorem}[Basic theorem of field extensions]
503 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 $a(x)$.
504\end{theorem}
505
506Let $a(x) \in F[x]$ be a polynomial of degree $n$. There is an extension field $E/F$ which contains all $n$ roots of $a(x)$.
507
508\section{Vector spaces}
509
510\begin{definition}[Vector space]
511 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
512
513 \begin{itemize}
514 \item $V$ with vector addition is an abelian group
515 \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:
516 $\forall k, l \in F,~\overrightarrow{a},\overrightarrow{b} \in V$
517 \begin{enumerate}[i.]
518 \item $k(\overrightarrow{a} + \overrightarrow{b}) = k\overrightarrow{a} + k\overrightarrow{b}$
519 \item $(k + l)\overrightarrow{a} = k\overrightarrow{a} + k\overrightarrow{b}$
520 \item $k(l\overrightarrow{a}) = (kl)\overrightarrow{a}$
521 \item $1 \overrightarrow{a} = \overrightarrow{a}$
522 \end{enumerate}
523 \end{itemize}
524\end{definition}
525
526\begin{definition}[Linear combination]
527 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
528 $$k_1 \overrightarrow{a_1} + k_2 \overrightarrow{a_2} + \cdots + k_n \overrightarrow{a_n}$$
529 is called a \emph{linear combination} of $\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}$.
530
531 The set of all the linear combinations of $\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}$ is a \emph{subspace of} $V$.
532\end{definition}
533
534\begin{definition}[Linear dependancy]
535 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 $k_1 \overrightarrow{a_1} + k_2 \overrightarrow{a_2} + \cdots + k_n \overrightarrow{a_n} = 0$.
536 Which is equivalent to saying that at least one of the vectors in $S$ is a linear combination of the others.
537
538 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 $k_1 = k_2 = \cdots = k_n =0$.
539 Which is equivalent to saying thatno vector in $S$ is equal to a linear combination of the other vectors in $S$.
540\end{definition}
541
542If $\{ \overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n} \}$ is linearly dependent, then some $a_i$ is a linear combination of the preceding ones.
543
544If $\{ \overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n} \}$ 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 $V$.
545
546
547\begin{theorem}
548 Any two bases of a vector space $V$ have the same number of elements.
549
550 (This comes from the fact that all bases of $\mathbb{R}^n$ contain exactly $n$ vectors)
551\end{theorem}
552
553If the set $(\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n})$ spans $V$, it contains a basis of $V$.
554
555\framebox{WIP: covered until chapter 28, work in progress.}
556
557\end{document}