Final

Question 1

Describe the Properties of a Ring

Answer 1

A set with 2 binary operations:
Abelian Addition and
Associative multiplication
Distributive property

Question 2

Cross products and rings?

Answer 2

Yes they exist and are obvious

Question 3

Theorem describing properties of negatives

Answer 3

1. a(-b) = (-a)b = -ab

2. (-a)(-b) = ab

Question 4

What is a ring with unity?

Answer 4

A multiplication identity exists

Question 5

What is a unit?

Answer 5

An element with an inverse

Question 6

What is a field?

Answer 6

A commutative ring where all elements have inverses

Question 7

True or false, every finite division ring is a field

Answer 7

True

Question 8

What is an integral domain?

Answer 8

A commutative ring with unity that has no zero divisors

Question 9

Which fields are integral domains?

Answer 9

All of them

Question 10

What is a zero divisor?

Answer 10

Elements not relatively prime to Zn

an element where ab does not = 0 for any b

Question 11

In a Quaternion, what is i^2 = j^2 = k^2

Answer 11

-1

Question 12

Describe the format of the quaternion equations

Answer 12

a1 + a2i + a3j + a4k

Question 13

In a Quaternion, what is ij, jk, ki equal to?

Answer 13

Multiplying the successive letter causes the next one.

ij = k, jk = I, ki = j

Question 14

In a quaternion, what is ji, kj, ik?

Answer 14

Multiplying the previous letter gives the negative of the next.
ji = -k, kj = -i, ik = -j

Question 15

When do the cancel laws hold?

Ex. ab = ac then b = c

Answer 15

When there is a ring with no zero divisors (iff)

Question 16

What is the characteristic of a ring?

Answer 16

The smallest n such that n*a = 0 for ALL a. Is 0 if no exists

Question 17

Characteristic of an integral domain?

Answer 17

A prime or 0

Question 18

Given a ring WITH UNITY, theorem that helps identify characteristic?

Answer 18

if n*1 not = 0 for all n, Char(R) = 0

if n*1 = 0 for some n, Char(R) = smallest such n

Question 19

Describe Fermat’s theorem:

Answer 19

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

Question 20

Describe an easy group you can form from fermats theorem.

Answer 20

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

Question 21

When to use fermat’s theorem?

Answer 21

What you are given a ring Zp where p is a prime

Question 22

What is Euler’s theorem?

Answer 22

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.

Question 23

Given Zn, how to tell how many units there are?

Answer 23

All the units are relatively prime to n.

in general, a unit is a non-zero divisor

Question 24

What is the euler phi function for n in Z?

Answer 24

Number of elements relatively prime to n

Question 25

What is R[X]?

Answer 25

The ring of polynomials. It is specifically a ring of the set of all polynomials of the finite SUM(ai*x^i)

Question 26

What is the degree of a polynomial?

Answer 26

The highest power

Question 27

What is the degree for the 0 polynomial?

Answer 27

It is undefined

Question 28

How to include multiple variables in a ring polynomial?

Answer 28

Let your ring be a ring polynomial.
Ex. Let R= Z5[X]
R[Y] = Z5[X,Y]

Question 29

Which properties are preserved when going to a ring polynomial?

Answer 29

All of them the original ring had

Question 30

If R is an integral domain, how to find units of R[X]?

Answer 30

They are the same units in the original ring R

Question 31

What is a simple group?

Answer 31

A Group that has no proper subgroups

Question 32

When is a finite abelian group simple?

Answer 32

IFF order of G is prime

Question 33

When is the alternating group simple?

Answer 33

When n>= 5

Question 34

Are normal subgroups preserved through homomorphism?

Answer 34

Yes, if N is normal subgroup, so is phi(N)

Same way in reverse

Question 35

How to decide if M is a maximal normal subgroup of G?

Answer 35

Show G/M is simple (IFF)

Question 36

What is the center of a group defined as?

Answer 36

Defined by the normal subgroup
Z(G) = {z in G | zg =gz for all g}
Elements that are multiplicative abelian for one other element

Question 37

What is the commutator normal subgroup?

Answer 37

Generated by c = {aba^(-1)b^(-1)} | a,b in G}

Question 38

How to show G is abelian with commutator and center?

Answer 38

G is abelian IFF Z(G) = G AND c = {e}

Question 39

Given a normal subgroup of G, how to show G/N is abelian?

Answer 39

IFF c is a subgroup of N