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Algebra 1, Definitionen-Manu
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Gravity
Terms in this set (112)
well-ordering
Every non-empty subset has a least element (smallest element-> smaller than every other)
a | b
a divides b -> b= c*a
gcd
greatest common divisor
a and b are
coprime
iff gcd(a,b)=1
integer p>= 2 is
prime
iff the only divisors of p are +-1, +-p
integer a>2 is
composite
if a is not prime
X,Y sets:
X x Y
Kartesian product-> X x Y = {(a,b)|a∊X, b∊Y}
involution
of a setX
is a map phi:X->X s.t. phi°phi = id_X
X set, R is a
relation
if R is a subset of X x X
(X,R) (set, relation) is a
partially ordered set
(poset)
if R satisfies:
- reflexivitiy: ∀ x: (x,x)∊R (also written xRx)
- anty-symm: ∀ x,y: if xRy and yRx -> x=y
- transitivity: if xRy and yRz -> xRz
(X,R) (set, relation) is a
totally ordered set
(toset)
if it is a poset and ∀ x and y: xRy or yRx
x∊X is a
upper bound
d* of Y
if ∀y∊Y : yRx holds
Chain
in poset (X,R)
∀ x,y in Chain xRy or yRx
max Element
of X
x∊X s.t if xRy -> y=x
S is
linearly independent
for all finite many vectors v1, ..., vn ∊S: A1 v1 +...+An vn =0 -> A1=....=An=0
S
spans
V
if ∀v∊V there exist finitely many v1, ...vn ∊S s.t v= A1 v1+...+An vn
S is
basis
of V
if S spans V and is linearly independent
Powerset
P(S)
Set of all subsets of S (-> has 2^|S| Elements)
Category C has theses 3 structures
-Obj(C) (some objects)
-for every 2 Obj(C) A and B: there is Hom(A,B) which is a set (an Element of this is called morpishm)
-Composition: Whenever A,B,C are objects of C Then (°composition)
°: Hom(A,B) x Hom(B,C) -> Hom(A,C) (=Hom(B,C)°Hom(A,B))
Category C satisfies Axioms
1) Every Hom(A,A) contains a distinguised element 1A ∊Hom(A,A) (called identity)
2) f°1A=f and 1B°f=f
3) Composition is associative: f,g,h: h°(g°f)=(h°g)°f
C category, f ∊Hom(A,B) a morphism. f is an
isomorphism
m* iff
iff there exist a g ∊Hom(B,A) s.t g°f=1A and f°g=1B
Covariant Functor consists of
Functor F: C->D , C,D being categories
1) Obj(C)->Obj(D), A->F(A)
2) ∀ morphism f∊Hom(A,B): T(f) is a morphism ∊Hom(T(A), T(B))
Contravariant Functor consists of
Functor F: C->D , C,D being categories
1) Obj(C)->Obj(D), A->F(A)
2) ∀ morphism f∊Hom(A,B): T(f) is a morphism ∊Hom(T(B), T(A))
Covariant Functor satiesfies
1) T(1A)=1 T(A)
2) T(f)°T(g)=T(f°g)
Contravariant Functor
1) T(1A)=1 T(A)
2) T(f)°T(g)=T(g°f)
Rings consits of
(R, +, *, 0, 1)
-R: Set
- + and * : R x R -> R
- 0, 1 ∊R
Ring-axioms
§Axioms for +
-Commutativity: a+b=b+a
-Associativity: (a+b)+c=a+(b+c)
-Zero: a+0=0+a=1
-Add. inv: ∀a∊R there is a b s.t a+b=b+a=0 (write: b=-a)
§Axioms for *
-Associativity: (ab)c=a(bc)
-Unit: 1 a=a 1=a
§Distributive Axioms:
-(a+b)*c = ac + ac
-a*(b+c) = ab + ac
Commutative ring
a b=b a
a, b in comm ring R are
associates
if there exists a unit u∊R with b=ua
R,S rings, function f:R->S is a
homomorphism
1) f(a+b)=f(a)+f(b)
2) f(a b) = f(a)*f(b)
3) F(1R)=1S (1R: unit in R) (F(0)=0)
ring R is
integral domain
1) R is commutative
2) 1 != 0
3) ∀ a,b∊R: a!=0 and ab=ac -> b=c (Cancelation of *)
equiv to 3) if x!=0 and y!=0 -> x*y!=0
a∊R unit
if there exist a c s.t a*c= 1R
Z_m
Z_m = Z/mZ= {0,1,2,...,m-1}
R ring, S∊R subset, S is a
subring
g*, iff
-0∊S
-1∊S
-S closed under +: a,b∊S-> a+b∊S
-S closed under mul: a,b∊S-> a*b∊S
-∀ a∊S: -a ∊S (add. inv)
Field
(F, +, *, 0, 1) if
-It is a commutative Ring
-0 != 1
-Every nonzero element is a unit (a∊F !=0 -> there is a b s.t a*b =1)
Field
(F, +, *, 0, 1) if (in therms of groups)
-(F, +, 0) is an abelian group
-(F*, multip , 1) is an abelian group
-Distributive Laws hold
F*
units
of F
F* = F \{0} (sometimes written with small x)
Field of Fractions
F
F={(a,b)|a,b∊R, b!=0} _ R
the set of equivalence classes under the Relation R: (a,b)R(c,d) iff ad = b*c
R[x], R ring
polynomial Ring:
R[x]= {an x^n +...+ a1 x + a0 | ai∊R}
R[[x]], R ring
powerseries in R
R[[x]]={a0 + a1 x + a2 x^2 + ... | ai∊R}
deg(f), f∊R[x]
n = deg(f) s. t an highest nonzero coefficient
F(x)
Field of Fraction of F[x]
F((x))
Field of Fraction of F[[x]]
F((x))={...a-2 x^-2 + a-1 x^-1 + a0 + a1 x^1 + a2 x^2 +....} with only
finitely
many negative powers
R comm ring, I ∊R ideal
-0∊I
-∀ a,b∊I -> a + b∊I
-∀ r∊R, a ∊I: r *a ∊ I
Equiv relation defined by I, ideal
a~b iff a-b∊I
Equivalence Relation R
-reflexivity: xRx
-symmetry: xRy -> yRx
-transitivity: xRy and yRz -> xRz
multiplicative
map
if f(xy)=f(x)f(y)
Ker(f), f hom
Ker(f) ={r | f(r) =0}
Im(f), f hom
Im(f) ={s| there exists r s.t f(r)=s}
Principle ideal (a)
(a)={x*a | x∊R}, R comm ring
Principle Ideal domain (PID)
R int domain, s.t every Ideal is of the form (a) for some a ∊R (every ideal is a principle ideal)
(a1, a2, ..., ak)
(a1, a2, ..., ak)={x1 a1 + x2 a2 + ... xk ak | xi ∊R} (is an ideal)
F field,
char(F)
Consider phi : Z -> F a ring homo ( phi(1)=1, phi(2) = 1+1...-> unique!)
-char(F)=0 iff Ker(phi)=(0) (-> |F| = oo)
-char(F)=p, iff Ker(phi)=(p), where p is prime
Fp (F: field F), p prime
Fp = Zp (Z: integ)
R comm ring, I ∊R
prime ideal
if R/I is a (int) domain
R comm ring, I∊R
maximal ideal
if R/I is a field
R comm ring, I, J Ideals: they are
coprime
iff I+J= R (eq: it exists a∊I, b∊J: a + b=1)
a commutative ring R is
local
if
there is an ideal I⊂R satisfying R*=R\I (I does not contain any units)
(this is equivalent to wanting I to be the unique maximal ideal)
(-> in local ring R\R^x is an ideal)
residue field
of a local ring R
is R/I for I being the ideal with R*=R\I
R comm ring, x!=0 ∊ R not a unit is called
irreducible
if x = a
b -> a or b in R
unit
R int domain:
unique factorization domain
(UFD)
Important: R int domain!
1)Every Element r∊R can be written as a product of irreducibles: r = p1 * p2...pn
2) Whenever p1 p2 ... pn = q1 q2 ... qm a product of irreducibles then m = n and there exist ei∊R unit , s.t pi = ei q _s(i), with s permutation
f∊R[x], R UFD,
content
t* of f: c(f)
c(f) = gcd (an, ..., a2, a1, a0) where
f = an x^n +... + a1x +a0
f∊R[x], R UFD: f*
such a poly that it holds: f = c(f) f*
f∊R[x] is
primitive
c(f)=1 (or unit)
Group (G, *, 1) consist of
-G :set
-*: G x G -> G
-1∊G (often also called e)
Group-axioms
-associativity: a(bc) = (ab)c
-unit 1a = a1= a
-inverse: ∀a∊G there exists a b s.t ab = ba = 1 (we call b = a^-1)
Abelian group
group that is commutative
index
of subgroup H in G
Gives the "relative size" of H in G (e.g. if H has index 2 in G, intuitavely half of the elements of G lie in H)
Symmetric group
Sn
Sn = {f:{1,....n}->{1,...n}| f is a bijection}
order
of a∊G
the minimal integer k s.t a^k = e (the unit)
if not existing: sometimes denoted as oo
G group, H is a
subgroup
- 1∊H
-h1, h2∊H -> h1*h2∊H
-h∊H -> h^-1∊H
homomorphism
of two groups
f: G1 -> G2 s.t ∀ x,y∊G1 it holds: f(x y)=f(x)*f(y) (with respective multiplication)
Ker(f)
for Hom
Ker(f)={x∊G1 | f(x)=1} (1 ∊G2)
Im(f)
for Hom
Im(f)={y ∊G2 | there exists a x ∊G1 s.t. f(x)=y}
two groups are
isomorphic
if there exist 2 Hom f:G1->G2 and g:G2->G1 s.t. f°g=1 and g°f=1
<a>
for a ∊G (group)
<a>={1,a,a^2,...,a^(k-1)} where k is the order of a (is a subgroup)
automorphism
of a group
a isom of a group to itself
centralizer
of g ∊G (group)
Centralizer C_G(g) = {x ∊G |x= g x g^-1} = {x ∊G | xg=gx}
center
of G (group)
Z(G)={g ∊G | xg=gx ∀ x ∊G}
for G group, x and y ∊ G are
conjugate
if there exists a g∊G s.t. g x g^-1 = y
conjugacy classes
of group
equivalence classes of relation: x~y iff x and y are conjugate
H is a
normal subgroup
of G
if ∀g∊G: gHg^-1=H. Denoted by H <| G (triangle)
quotient group
of G modulo H
If H <| G, the Group G/H is called quotient group
group G is
simple
if G ≠ {1} and the only normal subgroups of G are {1} and G
X ⊂ G subset of group G,
<X>
is the
subgroup generated by X
, i.e. the intersection of all subgroups H⊂G s.t. X⊂H.
Group action
is a: G x T-> T a map, s.t. (Notation: a(g,t)=g•t)
1) 1•t = t
2) g1 • (g2 • t)= (g1 g2) • t
faithful
Group-action
if for two different g,h∊G there exists a t∊T s.t. g • t ≠ h • t
or equivalent: for each g ≠ e in G there exists a t∊T s.t. g • t ≠ t
orbit
of G-action on T
Equivalence class of Relation ~, where x~y iff there exists g∊G s.t. g • x=y
Orb(x)
equivalence class of x ∊T with Relation ~, where x~y iff there exists g∊G s.t. g • x=y
action of G on T is
transitive
if T consists of a single orbit
fixed point
of G-action
x∊T is a fixed point of the G-action if ∀ g∊G: g • x=x.
In this case: Orb(x)={x}
stabilizer of x
: Stab(x)
stab(x)={g ∊ G| g • x=x} (is a subgroup of G)
a function f:T1->T2 is
G-equivariant
(G acts on T1 and T2)
if f(g • t1)= g • f(t1) ∀ g∊G and t1∊T1
r-cycle
element of Sn (a1, a2, ..., ar) (a1 goes to a2, a2 to a3...)
sign
of σ∊Sn
Say σ = (a1 b1)(a2 b2) ... (ak bk). Then: sign(σ)=even, if k≡0 (mod 2) (also 1), or uneven if k≡1 (mod 2) (also -1)
(sign(σ)=(-1)^k)
alternating group
An
An=Ker(sign)
field
extension
F field, F subfield of E- is the extension
(write E on top of F connected by |)
dim_F E ([E:F])
F ⊂ E ext of fields: then dim_F E ([E:F]) is defined to be the vectorspace dim of E as F-vector space
F⊂E arbitrary extension, α1, α2, ..., αk∊E some elements:
F(α1, α2, ..., αk)
F(α1, α2, ..., αk) is the smallest subfield of E which contains F & {α1, α2, ..., αk}
p poly
splitts completely
in F⊂E(ext)
if p(x)=(x-λ1)(x-λ2)...(x-λd), with λi∊E
The extension F⊂E(ext) is a
splitting field
d* for p over F if
-p splitts completely in E
-p does not splitt completely in ans subextension K: F⊂K⊂E (other then K=E)
isom of extensions
for F⊂E1 and F⊂E2 two extensions, ϕ isom is if fields E1->E2, s.t. ∀ λ ∊F: ϕ(λ)=λ
algebraic closure
extension F⊂E is alg closure of F, if
-every element e ∊E is algebraic over F (there exists a poly with root e)
-every poly q∊E[x] factors completely in E[x] (into lin. factors)
module
M over ring R (called
R-module
)
has two pieces of data:
1) (M, +, 0) is abelian group
2) there is a map •:RxM->M (scalar mult) that satisfies:
-r•(x+y)=r•x+r•y (r∊R, x,y∊M)
-(r+s)•x=r•x+s•x (r,s∊R, x∊M)
-r•(s•x)=(rs)•x
-1•x=x (1: id of ring)
hom of R-modules
f:M1->M2 that satisfies:
1) f is a hom of abelian groups
2) f(r•m)=r•f(m) (respects scalar mult)
M R-module is
finitely generated
iff there exist x1, ..., xk ∊M s.t. ∀ m∊M exist r1,...,rk: m=r1•x1+r2•x2+....+rk•xk
Tor(M)
, M module
Tor(M)={m∊M| there exists λ ≠ 0 , λ∊Z: λ•m = 0}
Free(M)
, M module
Free(M)=M/Tor(M)
cyclic
group
group generated by a single element -> <a> is a cyclic group
Ker(a)
a being a group action of G on T
Ker(a)={g∊G| g•t=t ∀ t∊T}
N_G(H)
, H⊂G subgroup
is the
normalizer
of H, and it is the stabilizer of the subgroup of H (->have this when working with group actions)
commutator
[a,b],
commutator subgroup
[G,G]
[a,b]=aba^-1b^-1
[G,G]={aba^-1b^-1, a, b∊G}
;