Question 1B Flashcards
(13 cards)
Theorem : Structure of finite abelian groups (pf not on exam)
Let G be a finite abelian group. Then there are positive integers
d_1, . . . , d_m with d_1 | · · · | d_m such that
G ∼= Z(bar)/d_1Z(bar) ⊕ · · · ⊕ Z(bar)/d_mZ(bar)
The integers d_1, . . . , d_m are uniquely determined by G.
Definition : Polynomial Ring
Given any ring R (think R = Z(bar) or R = C(bar)) and
variable X, the ring of polynomials over R is the set R[X] of all formal expressions
f = a_nX^n + a_(n−1)X^(n−1) + · · · + a_1X + a_0 (n ≥ 0, a_0, . . . , a_n ∈ R).
The adjective “formal” indicates that we are not viewing polynomials as functions, but rather just as expressions.
Definition : Direct Product
Given any two rings R and S, we can form their direct product
R × S = {(r, s) : r ∈ R, s ∈ S}
which is a ring with respect to the operations
(r, s) + (r′, s′) = (r + r′, s + s′)
(r, s) · (r′, s′) = (rr′, ss′)
and identities
0_(R×S) = (0_R, 0S)
1(R×S) = (1_R, 1_S)
Definition : Infinite product
Given a ring R we can form the infinite product of copies of R,
R^(N(bar)) = {(r_0, r_1, r_2, . . . ,) | r_i ∈ R} = {(r_i)(i∈N(bar))| r_i ∈ R}
which is a ring with respect to the operations
(r_i)(i∈N(bar)) + (s_i)(i∈N(bar)) = (r_i + s_i)(i∈N(bar))
(r_i)(i∈N(bar)) · (s_i)(i∈N(bar)) = (r_is_i)_(i∈N(bar))
and identities
0R^N(bar) = (0)(i∈N(bar))
1R^N(bar) = (1)(i∈N(bar)).
Also known as the ring of sequences with entries in R.
Definition : Formal power series
Given any ring R and variable X, the ring
of formal power series over R is the set R[|X|] of all formal expressions
R[|X|] = {SUM n∈N(bar) a_nX^n|a_n ∈ R}
Definition : Units
An element x ∈ R is said to be invertible if there is an element y ∈ R such that xy = yx = 1_R. Such an element is also called a unit, and its multiplicative inverse y is usually denoted x^(−1).
The set of all units is denoted R^× and called the group of units of R.
Definition : Group
A group G = (G, . , e) is a set G, a binary operation . : GXG -> G , and an element e epsilon G, S.T.
1) . IS ASSOCIATIVE : g . (h . k) = (g . h) . k for all g,h,k epsilon G
2) e is a unit for . : g . e = g = e . g for all g epsilon G
3) Every g epsilon G has an inverse : for g epsilon G there exists h epsilon G S.T. g . h = e = h . g
Lemma : The group of units R^× is a group under multiplication (pf not on exam)
For a ring R, the group of units R^× is a group under multiplication.
Definition : Division Ring (Skew Field)
A division ring (or skew field) is a ring R not equal {0} in which every non-zero
element x ∈ R is invertible.
Definition: K(bar) - algebra
Let K(bar) be a field. A K(bar) - algebra is a ring R S.T. K(bar) subset R and for all k epsilon K(bar) commutes with for all r epsilon R (kr=rk).
A K(bar) - algebra is a K(bar) - vector space.
Definition : Division algebra over K(bar)
A division algebra over K(bar) is a K(bar) - algebra that is a division ring.
Definition : Field
A field is a commutative division ring
Theorem : Frobenius (1877)
Let D be a finite-dimensional division algebra over a field K, where K is either C(bar) or R(bar).
(1) If K = C(bar) then D = C(bar).
(2) If K = R(bar) then D is isomorphic to R(bar), C(bar), or H(bar).
