MATH212 Flashcards
A ring is a non-empty set R with two binary operations, + and ·, satisfying the following conditions.
- (R, +) is an abelian group
- The operation · is associative
- The operation · is distributive over +
* **note that · does not need to be commutative in a ring
When is a ring commutative?
If · is commutative
When does a ring have an identity?
If · has an identity. This is usually denoted 1.
When does a ring have an inverse?
An element a in R has an inverse if it has an inverse with respect to ·.
We then call a a unit of R.
What is the definition of a subring?
Given a ring R, a non-empty subset S of R is a subring of R if, with the same operations for R, it itself if a ring.
S is a subring for R IFF for all a and b in S, both a-b and ab are in S
Let R be a ring. We say that R is a field if…
1) it has a multiplicative identity 1 != 0
2) it is commutative
3) every element other than 0 is a unit
What is a zero divisor?
A nonzero element a in R is a zero divisor if there is another nonzero element b in R such that ab=0
Suppose a commutative ring R has no zero divisors. What do you know about ab=ac if a!= 0
Then for any a!=0, b, c in R, b=c
What is an integral domain?
A ring R is an integral domain if it is commutative, has identity 1 != 0, and has no zero divisors
All fields are integral domains. All subrings of an integral domain which contain 1 are also integral domains.
What can you say about Z_n, n>=2
fields, zero divisors
If n is prime, Z_n is a field.
If n is not prime, Z_n contains zero divisors
What is the characteristic of an integral domain?
The order of its multiplicative identity in its additive group IF its finite, and zero if it is not finite.
What is the set S?
All ordered pairs in an integral domain R, whos second coordinate is not zero.
S = {(a,b) | a, b ∈ R, b != 0}
What is the set Q? (not rationals)
For any (a, b), (c, d) in S we define (a, b) ~ (c, d) if ad = bc, then ~ is an equivalence relation on S. We let [a, b] denote the equivalence class of (a, b) and we let Q denote the set of equivalence classes.
With the operations [a, b] + [c, d] = [ad + bc]
[a, b] · [c, d] = [ac, bd],
Q is a field.
What is a field of quotients?
The field Q with the operations [a, b] + [c, d] = [ad + bc]
[a, b] · [c, d] = [ac, bd]
For any (a, b), (c, d) in S we define (a, b) ~ (c, d) if ad = bc, then ~ is an equivalence relation on S. We let [a, b] denote the equivalence class of (a, b) and we let Q denote the set of equivalence classes.
An ordered ring is a commutative ring R with a subset R^+ satisfying …
1) R^+ is closed under addition
2) R^+ is closed under multiplication
3) if a is any element of R, then exactly one of the following hold: a in in R^+, -a is in R^+, or a = 0
What is an ordered field?
A field which is also an ordered ring.
Ordered rings are rings with a subset R^+ which is closed under addition, closed under multiplication, and for any element a of the ring a is in R^+, -a is in R^+, or a = 0
What is an upper bound for a set?
Let X be any set with a partial order <=.
If A is any subset of X, b is an upper bound for A if a<=b, for all a in A.
de Moivres Theorem:
Let z = r(cos(θ) + isin(θ)) be a non-zero complex number in polar form. For any positive integer n, we have z^n = r^n(cos(nθ) + i sin(nθ))
What is the identity of R[x]
The constant polynomial, f(x) = 1
When you multiply two polynomials f(x) and g(x), what is the degree?
deg(fg) = deg(f) + deg(g)
FACT: If R is an integral domain, then so is R[x].
If F is a field, then F[x]is an integral domain.
fact
Let R be an integral domain, how do you find the units in R[x]? What about fields?
The units in R[x] are exactly the constant polynomials f(x) = a, where a is a unit of R. If F is a field, then the units of F[x] are the non-zero constant polynomials.
Let F be a field and let f(x) and g(x) be in F[x] and assume g(x) != 0. There exist polynomials such that…
f(x) = q(x)g(x) + r(x), with either r(x) = 0 or deg(r) < deg (g)
What is an irreducible polynomial?
Let F be a field and f(x) be a polynomial over F with positive degree. We say that f(x) is reducible if it can be written as product of two polynomials, else is irreducible.
Reducing polynomials over integral domains:
A polynomial in an integral domain is irreducible IFF the polynomial is not a unit in the ring R[x]
FACT: Every polynomial of degree 1 in a field is irreducible
fact
FACT: Let f(x) be a polynomial over the field F and let a be an element of F. The polynomial (x-a) divides f(x) if and only if f(a) = 0.
fact
Rational roots theorem
Let f(x) be a polynomial a_n(x^n) + … + a_0. If p/q is any rational number, written so that p and q are relatively prime and a root of f(x), then p divides a_0 and q divides a_n
The Fundamental Theorem Of Algebra
Every polynomial of positive degree over C(omplex #) has a root
When is a polynomial in C[x] irreducible?
IFF its degree one
Let f(x) be a non-constant polynomial in R[x]. Then f(x) is irreducible IFF either
- f(x) is degree one
2. f(x) = ax^2 + bx + c, with b^2 - 4ac
What is a group?
1) The binary operation is associative
2) The set, G, has an identity for the operation
3) every element in G has an inverse (also in G) for the operation
Define the group U_n
U_n consists of all [k] in Z_n, where k is relatively prime to n.
U_n is an abelian group, with identity element 1
Define the set S_A, where A is a set
The set of all functions f : A → A which are bijective.
Deine S_n, where n is an integer
S_n denotes the set of all permutations of {1, …, n} with the binary operation of composition.
Has order n!
For n>=3, not abelian
What is a transposition?
A cycle of length 2
What is the sgn of a permutation?
-1 if odd
1 if even
FACT: jr^k = r^(n-k)j
fact
H is a subgroup of G IFF
- The identity is in H
- For all a, b in H, then ab is in H
- For all a in H, a^-1 is in H
OR/AND
- H is nonempty
- for all a, b in H, (a^-1)b is in H
FACT: All subgroups of cyclic groups are also cyclic
fact :)
Where G is a group, what is Tor(G)?
The torsion subgroup of G is all the elements in G with finite order
What is the ax + b group?
subgroup of GL_2(R), where
[ a b ]
[ 0 1 ]
a, b in R and a is not 0
What is the left coset of H in G?
gH = {gh | h in H}
What is the index of H in G?
The number of distinct left cosets, is denoted [G:H] = #G/H
Lagranges Theorem
o(G) = o(H)[G : H]
FACT: The order of any subgroup divides the order of the group
fact
What is the orbit of an element?
Let A be a nonempty set and let G be a subgroup of S_A. For any x in A, the orbit of x under G is:
G(x) = {g(x) | g in G}
What is the stabilizer of a group?
Let A be a nonempty set and let G be a subgroup of S_A. For any x in A, we define the set
G_x = {g in G | g(x) = x} which is called the stabilizer of x
Orbit Stabilizer Theorem
Let A be a nonempty set and let G be a subgroup of S_A. For any x in A, then for any x in A, we have #G(x) · o(G_x) = o(G)
Burnsides Lemma
A/G = 1/(#G) (the sum for all g in G) #A^g
What is a group isomorphism
A function f : G → H is called an isomorphism if
- f is a bijection
- f(ab) = f(a)f(b)
AXIOMS ON Z
- +
- has identity, associative, commutative, inverse - ·
- identity, associative, commutative - distributes
- Z^+
- closed for addition, multiplication, and a is in Z^+, -a in Z^+ or a=0 - Well ordered
