Final Flashcards
Describe the Properties of a Ring
A set with 2 binary operations:
Abelian Addition and
Associative multiplication
Distributive property
Cross products and rings?
Yes they exist and are obvious
Theorem describing properties of negatives
- a(-b) = (-a)b = -ab
2. (-a)(-b) = ab
What is a ring with unity?
A multiplication identity exists
What is a unit?
An element with an inverse
What is a field?
A commutative ring where all elements have inverses
True or false, every finite division ring is a field
True
What is an integral domain?
A commutative ring with unity that has no zero divisors
Which fields are integral domains?
All of them
What is a zero divisor?
Elements not relatively prime to Zn
an element where ab does not = 0 for any b
In a Quaternion, what is i^2 = j^2 = k^2
-1
Describe the format of the quaternion equations
a1 + a2i + a3j + a4k
In a Quaternion, what is ij, jk, ki equal to?
Multiplying the successive letter causes the next one.
ij = k, jk = I, ki = j
In a quaternion, what is ji, kj, ik?
Multiplying the previous letter gives the negative of the next.
ji = -k, kj = -i, ik = -j
When do the cancel laws hold?
Ex. ab = ac then b = c
When there is a ring with no zero divisors (iff)
What is the characteristic of a ring?
The smallest n such that n*a = 0 for ALL a. Is 0 if no exists
Characteristic of an integral domain?
A prime or 0
Given a ring WITH UNITY, theorem that helps identify characteristic?
if n*1 not = 0 for all n, Char(R) = 0
if n*1 = 0 for some n, Char(R) = smallest such n
Describe Fermat’s theorem:
Let p be a prime that doesn’t divide a in Z. Then p divides a^(p-1) - 1 or put better, a^(p-1) = 1 mod p
Describe an easy group you can form from fermats theorem.
Form a multiplicative group by removing all elements that do not have an inverse.
Group has order p-1 and all elements follow fermat’s theorem
When to use fermat’s theorem?
What you are given a ring Zp where p is a prime
What is Euler’s theorem?
More general version of fermat’s.
Let R be a ring with unity. Let U be the set of units.
Then U under multiplication is a group.
Given Zn, how to tell how many units there are?
All the units are relatively prime to n.
in general, a unit is a non-zero divisor
What is the euler phi function for n in Z?
Number of elements relatively prime to n
What is R[X]?
The ring of polynomials. It is specifically a ring of the set of all polynomials of the finite SUM(ai*x^i)
What is the degree of a polynomial?
The highest power
What is the degree for the 0 polynomial?
It is undefined
How to include multiple variables in a ring polynomial?
Let your ring be a ring polynomial.
Ex. Let R= Z5[X]
R[Y] = Z5[X,Y]
Which properties are preserved when going to a ring polynomial?
All of them the original ring had
If R is an integral domain, how to find units of R[X]?
They are the same units in the original ring R
What is a simple group?
A Group that has no proper subgroups
When is a finite abelian group simple?
IFF order of G is prime
When is the alternating group simple?
When n>= 5
Are normal subgroups preserved through homomorphism?
Yes, if N is normal subgroup, so is phi(N)
Same way in reverse
How to decide if M is a maximal normal subgroup of G?
Show G/M is simple (IFF)
What is the center of a group defined as?
Defined by the normal subgroup
Z(G) = {z in G | zg =gz for all g}
Elements that are multiplicative abelian for one other element
What is the commutator normal subgroup?
Generated by c = {aba^(-1)b^(-1)} | a,b in G}
How to show G is abelian with commutator and center?
G is abelian IFF Z(G) = G AND c = {e}
Given a normal subgroup of G, how to show G/N is abelian?
IFF c is a subgroup of N
