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Math 212 > MATH212 > Flashcards

MATH212 Flashcards

1
Q

A ring is a non-empty set R with two binary operations, + and ·, satisfying the following conditions.

A
  1. (R, +) is an abelian group
  2. The operation · is associative
  3. The operation · is distributive over +
    * **note that · does not need to be commutative in a ring
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2
Q

When is a ring commutative?

A

If · is commutative

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3
Q

When does a ring have an identity?

A

If · has an identity. This is usually denoted 1.

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4
Q

When does a ring have an inverse?

A

An element a in R has an inverse if it has an inverse with respect to ·.

We then call a a unit of R.

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5
Q

What is the definition of a subring?

A

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

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6
Q

Let R be a ring. We say that R is a field if…

A

1) it has a multiplicative identity 1 != 0
2) it is commutative
3) every element other than 0 is a unit

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7
Q

What is a zero divisor?

A

A nonzero element a in R is a zero divisor if there is another nonzero element b in R such that ab=0

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8
Q

Suppose a commutative ring R has no zero divisors. What do you know about ab=ac if a!= 0

A

Then for any a!=0, b, c in R, b=c

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9
Q

What is an integral domain?

A

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.

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10
Q

What can you say about Z_n, n>=2

fields, zero divisors

A

If n is prime, Z_n is a field.

If n is not prime, Z_n contains zero divisors

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11
Q

What is the characteristic of an integral domain?

A

The order of its multiplicative identity in its additive group IF its finite, and zero if it is not finite.

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12
Q

What is the set S?

A

All ordered pairs in an integral domain R, whos second coordinate is not zero.

S = {(a,b) | a, b ∈ R, b != 0}

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13
Q

What is the set Q? (not rationals)

A

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.

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14
Q

What is a field of quotients?

A

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.

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15
Q

An ordered ring is a commutative ring R with a subset R^+ satisfying …

A

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

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16
Q

What is an ordered field?

A

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

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17
Q

What is an upper bound for a set?

A

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.

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18
Q

de Moivres Theorem:

A

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))

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19
Q

What is the identity of R[x]

A

The constant polynomial, f(x) = 1

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20
Q

When you multiply two polynomials f(x) and g(x), what is the degree?

A

deg(fg) = deg(f) + deg(g)

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21
Q

FACT: If R is an integral domain, then so is R[x].

If F is a field, then F[x]is an integral domain.

A

fact

22
Q

Let R be an integral domain, how do you find the units in R[x]? What about fields?

A

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.

23
Q

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…

A

f(x) = q(x)g(x) + r(x), with either r(x) = 0 or deg(r) < deg (g)

24
Q

What is an irreducible polynomial?

A

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.

25
Q

Reducing polynomials over integral domains:

A

A polynomial in an integral domain is irreducible IFF the polynomial is not a unit in the ring R[x]

26
Q

FACT: Every polynomial of degree 1 in a field is irreducible

A

fact

27
Q

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.

A

fact

28
Q

Rational roots theorem

A

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

29
Q

The Fundamental Theorem Of Algebra

A

Every polynomial of positive degree over C(omplex #) has a root

30
Q

When is a polynomial in C[x] irreducible?

A

IFF its degree one

31
Q

Let f(x) be a non-constant polynomial in R[x]. Then f(x) is irreducible IFF either

A
  1. f(x) is degree one

2. f(x) = ax^2 + bx + c, with b^2 - 4ac

32
Q

What is a group?

A

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

33
Q

Define the group U_n

A

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

34
Q

Define the set S_A, where A is a set

A

The set of all functions f : A → A which are bijective.

35
Q

Deine S_n, where n is an integer

A

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

36
Q

What is a transposition?

A

A cycle of length 2

37
Q

What is the sgn of a permutation?

A

-1 if odd

1 if even

38
Q

FACT: jr^k = r^(n-k)j

A

fact

39
Q

H is a subgroup of G IFF

A
  1. The identity is in H
  2. For all a, b in H, then ab is in H
  3. For all a in H, a^-1 is in H

OR/AND

  1. H is nonempty
  2. for all a, b in H, (a^-1)b is in H
40
Q

FACT: All subgroups of cyclic groups are also cyclic

A

fact :)

41
Q

Where G is a group, what is Tor(G)?

A

The torsion subgroup of G is all the elements in G with finite order

42
Q

What is the ax + b group?

A

subgroup of GL_2(R), where
[ a b ]
[ 0 1 ]
a, b in R and a is not 0

43
Q

What is the left coset of H in G?

A

gH = {gh | h in H}

44
Q

What is the index of H in G?

A

The number of distinct left cosets, is denoted [G:H] = #G/H

45
Q

Lagranges Theorem

A

o(G) = o(H)[G : H]

46
Q

FACT: The order of any subgroup divides the order of the group

A

fact

47
Q

What is the orbit of an element?

A

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}

48
Q

What is the stabilizer of a group?

A

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

49
Q

Orbit Stabilizer Theorem

A

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)

50
Q

Burnsides Lemma

A

A/G = 1/(#G) (the sum for all g in G) #A^g

51
Q

What is a group isomorphism

A

A function f : G → H is called an isomorphism if

  1. f is a bijection
  2. f(ab) = f(a)f(b)
52
Q

AXIOMS ON Z

A
  1. +
    - has identity, associative, commutative, inverse
  2. ·
    - identity, associative, commutative
  3. distributes
  4. Z^+
    - closed for addition, multiplication, and a is in Z^+, -a in Z^+ or a=0
  5. Well ordered

Math 212 (3 decks)

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