Some Basics in Abstract Algebra

Before getting into origami, we need to develop a set of definitions needed to
understand the algebra in Auckly and Cleveland's paper.

2.1 Groups

Definition 2.1. A group is a set G together with a multiplication on G which
satisfies three axioms:

a) The multiplication is associative, that is to say (xy)z = x(yz) for any
three (not necessarily distinct) elements from G.

b) There is an element e in G, called an identity element, such that xe =
x = ex for every x in G.

c) Each element x of G has a (so-called) inverse x ^-1 which belongs to the
set G and satisfies x ^-1x = e = xx ^-1.

Definition 2.2. An abelian group is a group G such that for all x, y ∈ G,
xy = yx. (In this case, xy has an invisible operator , which could either be x+y
or x × y, but not both at the same time).

Definition 2.3. A symmetric group, , is the set of the permutations of n
elements .

Let us look at an example of symmetric group. List notation is used to
describe a set, , where are the elements of the
set. Using list notation, we can ex press the symmetric group of the elements




This set is consisting of all permutations of .

2.2 Ring

Definition 2.4. A ring R is a set, whose objects can be added and multiplied,
(i.e. we are given associations (x, y) → x + y and (x, y) → xy from pairs of
elements of R, into R), satisfies the fol lowing conditions : [4]

a) Under addition, R is an additive, and abelian group.

b) For all x, y, z ∈ R, we have

x(y + z) = xy + xz and (y + z)x = yx + zx

c) For all x, y, z ∈ R, we have associativity (xy)z = x(yz).

d) There exists an element e ∈ R such that ex = xe = x for all x ∈ R, where
e is the identity element.

An example of a ring is the set of integers, Z, because addition is commu-
tative and associative, and multiplication is associative. For any three integers
x, y, z, we have x(y +z) = xy +xz and (y +z)x = yx+zx. In addition, let the
multiplicative identity e = 1, then . Therefore, the set
of integers form a ring.

2.3 Field

Definition 2.5. A commutative ring such that the subset of non zero elements
form a group under multiplication is called a field.

A field is essentially a ring that allows multiplication to be commutative, after
removing the zero element. For a field, everything other than the zero element
must have an inverse. Otherwise it is a ring. An example would be that the set
of integers, Z, is not a field, but the set of rationals, Q, is a field. The reason is
that integers do not have multiplicative inverse: but . Matrices
are not a ring or a field, because it is not commutative under multiplication.

2.4 Polynomials

Definition 2.6. A number is an algebraic number if it is a root of a poly -
nomial with rational coefficients .

Definition 2.7. A polynomial p (x) in any field, F [x], is called irreducible
over F if it is of degree ≥1, and given a factorization, p(x) = f(x)g(x), with
f, g ∈ F [x], then deg f or deg g = 0.

For example, consider the following polynomials:



is reducible over Z[x] because both x + 2 and x - 2 are polynomials over
integers as well. However, is not reducible over Z[x]. It is, nonetheless,
reducible over Q[x], because both of the factored polynomials are polynomials
over Q.

Remark: Any algebraic number, α, could be expressed as a root of a unique
irreducible polynomial in Q[x], denoted by p_α (x). This polynomial p_α (x) will
divide any polynomial in Q[x] that has α as a root.

Definition 2.8. The conjugates of are the roots of the polynomial p_α (x).
An algebraic number is totally real if all of its conjugates are real. We denote
the set of totally real numbers by .

To make sense of the previous section, consider the number, . It is an
algebraic number, since it could be expressed as a unique irreducible polynomial
in Q[x]. The other roots of this polynomials are its conjugates: .
We can see that these numbers form a unique polynomial in Q:



which is a polynomial with rational coefficients. Additionally, since all the
conjugates of are real, is totally real. Additionally, the
polynomial above, p = x⁴ - 10x² + 17, is the unique irreducible polynomial,
which will divide any polynomials in Q[x] that contains as its root.
On the other hand, the number is not totally real, because two of its
conjugates,  , are imaginary.

The last topic we will go over is symmetric polynomials. We have already
defined symmetric group. The following is definition for symmetric polynomi-
als:

Definition 2.9. Let σ be a ring and let be algebraically independent
elements over R. Let x be a variable , and let G be a symmetric group on n
letters. Let σ be a permutation of integers (). Given a polynomial
, we define σf to be:



A polynomial is called symmetric if σf = f for all σ ∈ G.

For example, let . This is not a symmetric polynomial, because
we can let σ: :



However, is a symmetric polynomial, because .

Knowing the definition for symmetric polynomial, let's take a look at the fol-
lowing polynomial:



where each is given as:



The polynomials, , 1≤ j ≤ n are called the elementary symmetric
polynomials of .

Another way to express that is:



For example, expanding , we have:



All the following polynomials,



are elementary symmetric polynomials.

Now we are done introducing the definitions in abstract algebra that would
occur in this discussion of the paper folding. We can now start looking at some
properties of origami .