Algebra I¶

http://math.ntnu.edu.tw/~li/algebra-html/ https://www.youtube.com/watch?v=8qkfW35AqrQ&ab_channel=ProfessorMacauley

Preliminary and Basic Definition¶

01 Set.pdf

02 Equivalence Relations.pdf

03 Roots of unity.pdf

04 Binary Operations.pdf

05 Isomorphic Binary Structures.pdf

06 Strucutre Properties.pdf

07 Groups.pdf

Elementary properties¶

08 Elementary Properties of Groups.pdf

09 Groups of small orders.pdf

10 Multiplicative group modulo n.pdf

11 Subgroups.pdf

Cyclic Groups (Zn, Z)¶

12 Cyclic Subgroup.pdf

13 Cyclic Groups.pdf

14 Subgroups of Cyclic Groups.pdf
- is cyclic(remember?)

Permutation and Dihedral Groups¶

15 Group of Permutation.pdf
16 Cycle notations.pdf
17 Symmetry group of n-gons.pdf
- obit - stablizer to be more understood
19 Transpositions.pdf
- construction
  - adjacent -> non-adjacent
    - \((i, i + k) = (i + k, i + k − 1)(i, i + k − 1)(i + k, i + k − 1)\)
  - non-adjacent -> cyclic
    - \((a_1, a_2,..., a_n) = (a_1, a_n)(a_1, a_{n−1}). . .(a_1, a_2)\)
- inversions
- Even/Odd permutations
- Sign representations
  - Determinant
- Alternation Group

Cosets / Cayley Theorem / Lagrange Theorem¶

Key¶

Cancellation
- existance of inversion!!!!!

Notes¶

18 Cayley Theorem.pdf
- group are subgroup of \(S_{|G|}\)
- pf: see g as permutation and verify
20 Cosets.pdf
- coset properties are important
- what can do with the notation (by definition is obvious)
  - \((ab)H = a(bH) \not = aHbH\)
  - \(aH^2 = aH\) :::spoiler :::

21 Theorem of Lagrange.pdf (the lemma is actually more useful)

22 Applications of Theorem of Lagrange.pdf

Structure of Finitely Generated Abelian Groups¶

23 Diret Product.pdf

24 Finitely Generated Abelian Groups.pdf

25 Determine the Strucutres of Abelian Groups.pdf

Normal Subgroup¶

Overview¶

Kernel of Homomorphism
- must important concept (can ignore in preimage)
- \(gN = Ng = \phi^{-1}(\{\phi(g)\})\)
Definition
Properties
Index 2 (bipartition)
Quotient Group
- well defined by \(aHbH = abHH\)
First Isomorphism Thoerem
- by def is obvious

Detail¶

26 Homomorphism.pdf
- definition (non-trivial => not all to e)
- examples
  - evaluate homo
- isomorphisms
- ker?
27 Kernel of Homomorphism.pdf
- Key, by homomorphism and identity, all soubld be = obvious and trivial
- what maps to \(a \in G\) by homomorphism \(\phi :G \to G^{\prime}\) is \(aH = Ha\) where \(H = ker(\phi)\) :::spoiler :::
28 Normal Subgroups.pdf
- definition
  - \(gN = Ng ~\forall g \in G\)
  - explicitly note that does not means each element is the same
    - (consider dihedral)
- proposition: subgroup of index 2 (partition is binary)
- properties
  - pf:
- quotient group
  - set of cosets with well-defined op
    - requires normal to let \(aHbH\) be \(abH\)
  - \((G/N, ∗)\) is a group, called the quotient group of \(G\) by \(N\)
- - my comment: the recall that here should be "by definition"
- First Isomorphism Theorem
  - G/N is group since N is ker(h) thus normal
  - example of application
29 Quotient Group Computation.pdf
- want to calc with familiar iso groups
  - pf: by FIT
- elementary divisor theorem
  - pf: TB understood
  - view basis as linear comb of original
  - relation matrix
    - \((a_{ij})n, m\) (n: basis as l.c., m: basis of kernel)
  - usage

Exam 1 (97/105)¶

2020/11/5 linearity's power(range) - wrong at - show \(A_4 \not = D_6\) - no order-6 element in A4 - -8 in total - TB more discussed - [ ] orbit-stablizer theorem - [ ] proof of the foundamental theorem of finitely generated abelian groups - [ ] classification of finite groups - [ ] elementary divisor theorem

Normal+¶

invariant through homomorphism¶

note that N to G \(\iff\) \(\phi(N)\) to \(\phi(G)\)

Center \(Z(G)\)¶

\(\{x \in G \mid xg=gx~~\forall g \in G\}\)

Commutator Subgroup \([G, G]\)¶

\(\{a^{-1}b^{-1}ab \mid a, b \in G\}\)

Theorem¶

:::success \([G, G]\triangleleft G\). \(G/N\) (s.t. \(N\triangleleft G\)) is abelian \(\iff [G, G] < N\). :::

:::info 1.\([G,G]\triangleleft G\) by checking criterion \(g^{-1}Ng \in N\) 2. \(G/N\) is abelian \(\iff (aN)(bN)=(bN)(aN)\) for all \(a,b\) \(\iff abNN=baNN\) for all \(a,b\) (N normal) \(\iff [G, G] < N\) 2. actually obvious by the meaning of quotient is "see as same" :::

Normal Core¶

recall: normal iff invariant under all conjunctions (\(g^{-1}Ng=N\))
consider \(\bigcap_{g \in G}g^{-1}Hg\) for some subgroup \(H\)
- invariant under conjunction
- still subgroup of \(G\) and of course \(H\) by intersection
- cell this the normal core of \(H\)
- which is the largest normal subgroup in \(H\)

Graph Automorphisms and Caley Graphs¶

Group of Graph Automorphisms¶

\(G = (V, E)\)
definition of \(Aut(G)\) (a group under composition)
- \(\{f: V \to V \mid (x, y) \in E \implies (f(x), f(y)) \in E \}\)

Caley Graph¶

A graph given by \((G, S)\)
- \(G\) is a group
- generating set \(S \ni s \in S \implies s^{-1} \in S\)
- \((a, b) \in E \text{ if } as = b \text{ for some } s \in S\)

A Theorem¶

Ring and Field¶

Key¶

additional structure for ring
- unity
  - unit
- commutation

Note¶

Ring
- Abelian Group
  - \((R, +)\)
- Multiplication
  - associative, \((ab)c = a(bc)\)
- Distribution Law
  - \((a+b)c = ac + bc\)
Ring Properties
- \(0 \cdot a = 0\)
- \(a(-b) = -ab\)
- these are proven by distribution law
Ring+
- commutative
  - multiplication satisfies \(ab = ba\)
- with unity
  - multiplication idendity
  - i.e. \(\exists e \ni ae = e \forall a \in R\)
- unit (defined if ring is with unity)
  - element that has inverse
  - property: the set of units form a group
- division ring (skew field)
  - every element is unit
- field
  - commutative division ring
subring is easy to check as subgroup
homomorphism and isomorphism is defined as usual
example
- \(\mathbb{C} \cong \{ (a, b, -b, a) \mid a, b \in \mathbb{R}\}\) (matrix``)

Integral Domain¶

Key¶

cancellation law of multiplication \(\iff\) no zero divisors

Notes¶

motivation
- solution to \(ab = 0\) is not necessarily \(a = 0\) or \(b=0\)
divisors of zero
- def:
  - \(a \not = 0 \in \mathbb{R} \ni \exists b \not = 0, ab=0\)
- examples
cancellation law of multiplication \(\iff\) no zero divisors
- hint: distibution law for the non-trivial direction
notation \(\frac ba\)
- used when
  - a is a unit (has inverse)
  - R is commutative
definition
- commutative ring
- with unity \(1 \not = 0\)
- no zero divisors
example
- direct product is always not integral domain
fields and integral domains
- fields are integral domains
  - \(0 = ab = a^{-1}ab = b\)
- finite integral domains are fields
  - pf: hint cancellation law
    - by cancellation, and list all elements in \(D\)
    - consider left product of an element a
    - by cancellation, all elements distinct + cardinality same
    - => must be one only \(a_i\) s.t. \(aa_i = 1\)
  - if infinite?
    - cannot use the arguement of distinct and number of elements are the same

Characteristic of a ring¶

definition
- the least positive integer s.t. \(n \cdot a = 0~\forall a\in R\)
- if no such \(n\), let it be 0
examples
- \(\mathbb{Z}\) is of characteristic \(0\)
- \(\mathbb Z_n\) is of characteristic \(n\)
theorem: check \(1\) if R is ring with unity \(1\)
- be careful that \(n\) is not in \(R\) here
- \(n\cdot a = a+a+a+a \dots = a(1+1+\dots+1) = a(n \cdot 1) = 0\)
theorem: let R be ring with unity if characteristic \(n\), \(\langle 1 \rangle \cong \mathbb{Z}/n\mathbb{Z}\)
QA:
- how to show \((m \cdot 1) \cdot (n \cdot 1) = (m \cdot n) \cdot 1\)
- distribution law + counting

Field of Quotient¶

extend (infinite) integral domain \(D\) to a field \(F\)
- recall the definition carefully
- i.e. we want inverse for all elements
add \(a^{-1}\) denoted as \(\frac{1}{a}\) and their multiplication and addition
\((a,b), (c,d)\) should be consider same if \(ad=bc\) (think as inverse)
so define a equivalent relation
Now \(D\) is (isomorphic to) a subdomain of \(F\)
one can show such F is the smallest field containing \(D\) (up to iso)

Ring of Polynomials¶

Ring of Polynomials II¶

The First Ring Isomorphism Theorem¶

http://math.ntnu.edu.tw/~li/algebra-html/node66.html

Ring Extension¶

Polynomial¶

Division Algorithm of Polynomials¶

given \(f(x), g(x)\)
\(f(x) = g(x)q(x) + r(x)\) for some unique \(q, r\) with \(deg(r) \lt deg(g)\)
proof hint:
- by considering the set \(S = \{ f(x)-g(x)s(x) \mid s(x) \in \mathbb{F}[x] \}\)
- consider r(x) tp be an element with miminal degree in \(S\)
- make use of \(\mathbb{F}\) has inverse
division algorithm holds for integral domain if leading term of \(g\) is invertible

Zeros of Polynomials¶

definition
- \(a \in \mathbb{F} \ni f(a) = 0\)
theorem
- \(f(a) = 0 \iff x − a\) is a factor of \(f(x)\)
- proof hint
  - division algorithm
theorem
- A nonzero polynomial \(f(x) \in \mathbb{F}[x]\) of degree \(n\) can have at most \(n\) distinct zeros in \(\mathbb F\).
- proof hint
  - by induction
  - care require fields are integral domains thus have no zero divisors
  - self thinking problem
    - multiplicity of zero and at most \(n\) zeros up to multiplicity
theorem
- \(G\) be a finite subgroup of the multiplicative group \(F^{\times}\) then \(G\) is cyclic
- proof hint
  - the fundamental theorem for finitely generated abelian groups
  - a subgroup \(H\) of \(G\) isomorphic to only once (by taking power to \(e_{i}\)?)
  - then has to be distinct else \(x^p=1\) has \(p^2\) solutions
- proof
(supplement) Wilson's Theorem
- \((p-1)!=-1 \mod p\)
- pf:
  - by inverse, and \(x^2=1 \iff x=\pm1\)
- corallary: \(x^p-p = \prod_i (x-i)\)

Irreducible Polynomials¶

definition
- if \(f(x) = g(x)h(x)\) for some \(g, h \ni deg(g) < deg(f)\) and \(deg(h)<deg(f)\), \(f(x)\) is reducible
- otherwise, \(f\) is irreducible (over \(\mathbb F\) if \(g, h\) are restricted in \(\mathbb{F}[x]\))
- equivelant definitions (0-deg functions are all units of \(\mathbb F[x]\))
if of degree 2 or 3
- \(f\) must has a zero since \(2 = 1+1\), \(3 = 1+2 = 1+1+1\)
- remark: \(4 = 2+2\)

Gauss's Lemma¶

note: calculations
example
- factorize \(x^4+1\) over \(\mathbb{Q}\) and \(\mathbb{Z}_3\)
- hint:
  - assume \(x^4 + 1 = (x^2+ax+b)(x^2+cx+d)\) and start with \(bd =1\)

Proof of Gauss's Lemma¶

content of \(f\)
- \(\mathrm {Con}(f) = gcd(a_0, a_1, \dots, a_n)\)
\(f\) primitive := \(\mathrm {Con}(f) = 1\)
lemma
- \(\mathrm {Con}(g) = \mathrm {Con}(h) = 1 \implies \mathrm {Con}(gh) = 1\)
- pf hint: by mod p = 0 and no zero divisors
proof:
- assume \(\mathrm {Con}(f) = 1\) is ok (diff is constant)
- assume \(f(x) = ag(x) \cdot bh(x)\) where \(\mathrm {Con}(g) = \mathrm {Con}(h) = 1\)
- show \(ab = \frac r s = 1\)
- \(s\mathrm {Con}(f) = \mathrm {Con}(sf) = \mathrm {Con}(rgh) = r\mathrm {Con}(gh) = r\)
corollary
- irreducible over \(\mathbb{Z} \iff\) irreducible over \(\mathbb{Q}\)
- zero of f is of form \(\frac b c \ni b \mid a_0, c\mid a_n\)

Factorization Modulo a Prime¶

reduction over \(m\) is a homomorphism
mod m irreducible \(\implies\) irreducible
- not true for another direction
- hint: homomorphism

Eisenstein criterion (!!)¶

\(f(x) \in \mathbb{Z}[x]\), \(p\) is a prime
if
- \(a_n \not = 0 \mod p\)
- \(a_i = 0 \mod p ~ \forall i <n\)
- \(a_0 \not = 0 \mod p^2\)
then
- \(f(x)\) is not reducible over \(\mathbb{Q}\)
pf hint
- claim the \(\mod p\) conditions give that \(\bar{f}=\bar{g}\bar{h}\) where \(\bar{f}, \bar{g}, \bar{h}\) are in the form \(\{a,b,c\}_kx^k\)
- then \(a_0 \not = 0 \mod p^2 \implies \bot\)
- pf of claim
  - let \(r\) be the least integer s.t. \(b_r \not = 0 \mod p\)
  - \(s\) be that for \(h(x)\)
  - then \(a_{r+s} = \sum_{i=0}^{r+s}b_ic_{r+s-i} = b_rc_s \not = 0 \mod p\)

Cyclotomic Polynomials¶

\(\phi_p(x) = \sum_{i=0}^{p-1}x^{i} = \frac{x^p-1}{x-1}\) is reducible over \(\mathbb{Q}\)
- pf hint
  - binomial expand \(\phi_p(x+1)\)
    - this step needs a lemma, pf?
  - constant term = \(p\)
  - Eisenstein Criterion
\(\phi_p(x)\) is called p-th cyclotomic polynomial
The zeros of it are precisely the p-th roots of unity, except for 1.
In general, n-th cyclotomic polynomial is
- \(\begin{align*} \phi_n(x) = \prod_{gcd(k,n)=1}(x-\zeta_n^k) = \prod_{a \in \mathbb{C} \mid ord(a)=n}(x-a) \end{align*}\)
- where \(\zeta_n^k=e^{\frac{2\pi i k}{n}}\)
self thinking
- show \(\phi_n(x)\) is always a integer polynomial.

GCD, Unique Factorization Theorem of F[x]¶

CD of polys (trivial definition)
GCD is unique up to units
\(af+bg=h\)
lemma: suppose \(f=gh\), \(p\) irreducible in \(F[x]\) and \(p|f\), then \(p|g\) or \(p|h\)
- pf by construction with \(af+bg=1\)
Unique Factorization Theorem (statement is intuitive)
- existence by induction
- uniqueness by lemma + cancellation law of integral domain (remain Q?)

Polynomial Ring Modulo A Polynomial¶

def: Ring of Residue Classes
- intuitive
thm: poly \(f\) is irreducible \(\iff\) \(\mathbb{F}_f[x]\) is a field
- hint:
  - to the left, suppose \(f = gh\), no zero divisor by fields are I.R.s
  - to the right, suppose \(f\) is irreducible, by gcd argument construct inverse for all \(g\)
thm: \(F_f[x] \cong \frac{F[x]}{f(x)F[x]}\)
- pf: the 1st ring homomorphism(remain Q?)

Field Extension¶

\(E\) is called a field extension of \(F\) if \(F \le E\)
if \(E=F(\alpha)\) for some \(\alpha \in E\) then \(E\) is called a simple extension of \(F\)
- note:
  - \(F[a] := \{f(a)\}\)
  - \(F(a) := \{f(a)/g(a)\}\)
def:
- algebraic or transcendental over \(F\)?
- exist \(f(x)\in F[x] \ni f(\alpha)=0\)?
thm:
- transcendental evaluation is a isomorphism from \(F[x] \to F[\alpha]\)
- proof is trivial

Irreducible poly for \(\alpha\) over \(F\)¶

def:
- \(I_{\alpha} = \{f(x) \in F[x] \mid f(\alpha)=0 \}\)
- \(p_\alpha\) be a non-zero poly in \(I_\alpha\) with minimal degree
thm: \(p_\alpha(x)\) is irreducible
- pf: let \(p=gh\), then \(g(\alpha)=0\) or \(h(\alpha) = 0\)
thm: \(I_{\alpha} = p_{\alpha}(x)F[x]\)
- pf: exercise
  - my guess, division algorithm
\(Irr(\alpha, F)\), the irreducible polynomial for \(\alpha\) over \(F\), the unique monic irreducible poly \(p(x) \in I_{\alpha}\)
\(deg(\alpha, F)\), the degree of \(\alpha\) over \(F\) \(:=deg(Irr(\alpha, F))\)

Basis of Extension Fields¶

keys
- by \(Irr(\alpha, F)\), GCD, Division
thm:
- let \(E=F(\alpha)\) be a simple algebraic extension of \(F\), \(F(\alpha) = F[\alpha]\)
- pf hint:
  - find "inverse" of \(g(\alpha)\) by GCD of \(Irr(\alpha, F), g\)
thm:
- \(\forall \beta \in F[\alpha], \beta\) can be written in \(F^{deg(Irr(\alpha, F))}[\alpha]\)
- pf hint: division algorithm
do some examples
- find solution of fraction by GCD algo, or observation
recap vector space
thm:
- \(F[\alpha]\) is a vector space with a basis \(\{1, \alpha, ..., \alpha^{n-1}\}\)
- for the case \(\alpha\) is transcendental, \(n=\infty\)

Algebraic Extension¶

- thm: finite extensions are algebraic extensions - pf hint: \(\{1, \alpha, \alpha^2 \dots, \alpha^n \}\) linear dependent => has non all zero... - thm: \([K:F] = [K:E][E:F]\) - pf hint: by def, expand twice - apps -

Algebraic Closure and Constructions of Field Extension¶

Thm: let \(E\) be a finite extension of \(F\), if \(\alpha, \beta\) are algebraic over \(F\), so is \(\alpha+\beta\) and \(\frac\alpha\beta\)
- pf:
  - its suffice to show \(F(\alpha, \beta) = F(\alpha)(\beta)\) is a finite extension of \(F\), which is obvious
  - then \([F(\alpha,\beta):F] = [F(\alpha,\beta):F(\beta)][F(\beta)):F]\)
  - we then know they are algebraic (finite=>algebraic)
  - key: 2 step extension
def:
- Algebraic Closure of F in E
  - \(\bar{F}_E=\{\alpha \in E \mid \alpha\text{ is algebraic over }F\}\)
  - is a subfield of E
- pf of is a field, by the above thm
def:
- Algebraically Closed
  - every non-constant polynomial in \(F[x]\) has a zero in \(F\)
  - remark: so every poly is split in such F
def:
- An algebraic closure \(\bar F\) of a field \(F\)
  - an algebraic extension of \(F\) that is algebraically closed.
  - remark: the maximal field one may obtain by algebraic means
- ex:
  - \(\mathbb{C}\) to \(\mathbb{R}\)
  - \(\bar{\mathbb{Q}}\) to \(\mathbb{Q}\)
pf of \(\bar{\mathbb{Q}}\) to \(\mathbb{Q}\)
thm: every field has algebraic closure

Construction of (Algebraic) Field Extension (the two ways)¶

quotient field
a bigger field's subfield
comparison

Finite Field¶

https://jupiter.math.nctu.edu.tw/~weng/courses/alg_2007/Algebra%202006-2/Handout-Section33.pdf http://math.ntnu.edu.tw/~li/galois-html/galois.pdf go through the exercise in winter vacation ! - propositions - - 1 is shown - 2 is trivial (hint: assume \(p=rs\)) - 3, 4 is direct proof (must contain \(n \cdot 1_F\), for Q, \((n\cdot 1_F) (m \cdot 1_F)^{-1}\)) - thm: finite field orders are \(p^n\) - by proposition 3 - how to explicitly construct a finite field of certain order? - - \(GF(p^n)\) and \(\phi_n(x)\) -

My Note for the scope of final exam¶

About field extension and finite field(\(F_{p^n}\))¶

algebraic extension and basis
- deg(irr)
- finite <=> algebraic
- deg of extension
  - \(n!\) by \(f\) to \(f/(x-\omega)\)
  - \(m \mid n\)
finite field assumptions
- group of unit of field is cyclic
- \(char(F) = p\) or \(0\)
- F is (iso to) extension of \(Z_p\) if \(char(F) = p\)
  - and 0
Link \(Z_p(\beta)\) to \(F_{p^n}\) (unique!)
- by \(x^{p^n}-x\)
- proof iff
  - cardinality if group of unit + in closure => by order arguement is zero of the function
  - zeros of the function =>
- show existence by derivative and no repeated root
- uniqueness
Cyclotomic polynomials over finite fields
Factorization of \(x^{p^n}-x\)

problem type¶

extension basis calculation
- get inverse
- represent zero of other poly
- matrix correspondence
definition and implications
show \(\phi_k(x)\) is irr. over \(Z_p\)
- by argument that
  - \(ord(\alpha)=k\)
  - want \(\alpha\) generate extension of full degree over p
  - \(\alpha \in F_{p^k}\) but not in divisors
uniqueness of finite field
matrix fields and their group of unities' structure
- use finitely generated subgroup + order argument
- final : (a, b, 0 a) => \(Z_p \times Z_{p-1}\)

Review¶

Group
- group structures
  - cyclic
  - commutative
  - cardinality
- cyclic groups
  - \(Z, Z_N\)
- finitely generated abelian group
  - \(\prod Z_{{p_i}^{e_i}} Z^n\)
- non-abelian group
  - \(S_n, A_n, D_n, Q_8\)
- subgroup
  - coset
    - Lagrange Thm
  - normal subgroup
    - as kernel of group homomorphism
    - induce quotient group
- simple group
  - 5 classes + 27 special case
- group action
  - burnside's lemma
    - \(|X/G| = \frac{1}{|G|}\sum_{g \in G}|X^g|.\,\)
  - sylow theorem (next semester)
    - existence of \(p\)-subgroups
    - criterion for normalness of \(p\)-subgroups
  - the isomorphism theorems
    - first (this semester)
Ring
- commutative (this semester)
  - integral domain
    - field
  - (not this semester)
- non-commutative (not in this semester)
  - Matrix Ring
Field
- extension
  - all fields are extensions of \(Z_p, Z\)
- finite field, \(char(F) = p\)
  - \(F_{p^n}\) is universal
    - final will be a problem to contruct \(|F| = 25, a+b\alpha\)
  - how to construct such fields
    - find deg=n, Z_p pily
- infinite field, \(char(F) = 0\)

next semester¶

group
- isomorphism theorems
ring (commutative)
- integral domain
  - PID
  - UFD
    - Fermat's Last Theorem
      - a showcase of "algebraic number thoery"
  - ED
- ideals
- CRT
- Fermat's Little Theorem
- RSA
  - CRT + Fermat's Little Theorem Application
ring (non-commutative)
- quaternion
Golais Theory
- study the group of symmetries of a field extension
- when an extension is Golais(include all zeros of a poly) => a bijection between subfields and subgroups
  - solving poly eq by radical

Last update: 2023-09-15