Definition of field Aim lecture: One usually does linear algebra over - - PowerPoint PPT Presentation

Definition of field

Aim lecture: One usually does linear algebra over some “system of numbers”, or more precisely, a field. We introduce this notion in full generality here.

Defn

A field consists of an additive (hence abelian) group (F, +) equipped with a second associative, commutative binary operation × called multiplication such that

1

× has an identity 1 = 0 called the field or multiplicative identity,

2

there exist inverses for non-zero elements wrt ×, that is, given β ∈ F −0, there is some β−1 ∈ F with β × β−1 = 1 = β−1 × β.

3

the following distributive law holds, for β, β′, β′′ ∈ F we have β × (β′ + β′′) = β × β′ + β × β′′. From now on, throughout these lectures, the symbol F will always represent a field

• f some sort. The addition is always denoted + but the multn will usually be

abbreviated to β × β′ = ββ′.

Daniel Chan (UNSW) Lecture 4: Fields Semester 2 2012 1 / 10

Examples

Daniel Chan (UNSW) Lecture 4: Fields Semester 2 2012 2 / 10

Multiplicative group

Prop-Defn

Let F be a field as usual.

1

For β, β′ ∈ F, we have ββ′ = 0 iff either β = 0 or β′ = 0.

2

Multn restricts to a binary operation on F× = F −0.

3

(F×, ×) is an abelian group called the multiplicative group of F.

• Proof. Note that 1) =

⇒ 2) whilst 3) follows from field axioms for × and 2) so we

• nly prove 1).

1) (⇐ =). 0 + 0β = 0β = (0 + 0)β = 0β + 0β so cancellation in the additive group (F, +) gives 0 = 0β. By commutativity of multn, see also β0 = 0. 1) (= ⇒) Suppose that ββ′ = 0 but β = 0. Picking an inverse β−1 to β & using associativity of multn we see β′ = β−1ββ′ = β−10 = 0 by the (⇐ =) part already

• proved. This completes the proof of the propn.

Rem Since F× is a group, we now know multiplicative inverses are unique & can use other facts about groups.

Daniel Chan (UNSW) Lecture 4: Fields Semester 2 2012 3 / 10

Basic properties

Note that in any abelian group (A, +) we can define subtraction by β − β′ = β + (−β′). In particular, we can subtract in any field & sim divide by non-zero elements. The field axioms ensure most of the usual arithmetic rules hold

Prop

Let F be a field as usual & β, β′, β′′ ∈ F.

1

(−1)β = −β

2

β(−β′) = −(ββ′)

3

β(β′ − β′′) = ββ′ − ββ′′

• Proof. Ex

Daniel Chan (UNSW) Lecture 4: Fields Semester 2 2012 4 / 10

Some finite fields

This section is not examinable as it depends on you having done MATH1081. It is to give you some interesting examples of fields. Let p be a prime & Fp = {0, 1, . . . , p − 1}. For β, β′ ∈ Fp, we consider the binary

• perations

β + β′ = (β + β′) mod p, ββ′ = (ββ′) mod p. It is fairly easy to see that these are commutative associative binary operations with identities. Furthermore, (Fp, +) is an abelian group. The existence of multiplicative inverses for β ∈ F −0 is harder & amounts to the fact that β has an inverse modulo p.

Theorem

Fp is a field with the above addn & multn.

• Proof. Omitted.

In any field, we may let n ∈ Z represent the element n1. In Fp we have p = 0!

Daniel Chan (UNSW) Lecture 4: Fields Semester 2 2012 5 / 10

Polynomials

A polynomial over (the field) F in the indeterminate x is an expression of the form p(x) =

• i≥0

pixi where every pi ∈ F & pi = 0 for i ≫ 0 i.e. there are only finitely many non-zero co-efficients pi. For such a polynomial you can define degree, leading co-efficient, leading term, constant term, term in the usual way. Also, we can write p(x) as p(x) = p0 + p1x + . . . + pdxd if pi = 0 for i > d. Further, we may omit terms with zero co-efficient. Let F[x] denote the set of all polynomials over F. It’s easy to prove

Prop

(F[x], +) is an abelian group if we define addition on F[x] co-efficient-wise by (

• i

pixi) + (

• i

qixi) =

• i

(pi + qi)xi. Note there is no clash in notn with p(x) = p0 + p1x + . . . + pdxd.

Daniel Chan (UNSW) Lecture 4: Fields Semester 2 2012 6 / 10

Polynomial arithmetic

We define multiplication on F[x] by (

• i

pixi)(

• j

qjxj) =

• k

(

• i+j=k

piqj)xk.

Prop

1

Multn on F[x] is associative & commutative & there is a multiplicative identity, namely, 1.

2

The distributive law holds i.e. for p(x), q(x), r(x) ∈ F[x] we have p(x)(q(x) + r(x)) = p(x)q(x) + p(x)r(x).

• Proof. Long ex.

Unfortunately, F[x] is never a field.

Daniel Chan (UNSW) Lecture 4: Fields Semester 2 2012 7 / 10

Field of rational functions on R & C

Let F = R or C. Then any polynomial p(x) =

i pixi ∈ F[x] can be viewed as a

function p(x) : F − → F : β →

i piβi. As you know, in this case, polynomial

addn & multn agree with usual pointwise addn & multn of functions. A rational function is a fraction of the form f (x) = p(x)

q(x) for some p(x), q(x) ∈ F[x]

with q(x) = 0. We will view this as a partially defined function from F − → F. In particular, we identify fractions if they agree as functions wherever they are both

• defined. Let F(x) denote the set of all such rational functions.

Prop

R(x) and C(x) are fields when endowed with pointwise addn & multn.

• Proof. Easy but long ex.

Rem In fact there are analogous fields of rational functions F(x) for any field, F,

Daniel Chan (UNSW) Lecture 4: Fields Semester 2 2012 8 / 10

Matrices

Most of the theory of matrices over R you learnt in 1st year carries over to matrices over F for any field F. In particular we have

Defn

Let (aij), (bij) ∈ Mmn(F), (cij) ∈ Mlm(F).

1

Matrix addn (aij) + (bij) = (aij + bij)ij

2

Matrix multn (cij)ij(ajk)jk = (

j cijajk)ik

We have the following facts as in 1st year with the same proofs. (Mmn(F), +) is an abelian group. Multn is associative & the identity matrix is a multiplicative identity. The distributive law holds whenever all matrix sums & products are defined.

Daniel Chan (UNSW) Lecture 4: Fields Semester 2 2012 9 / 10

Remarks on solving linear equations

Fields provide the proper context for solving linear eqns. For example, we may solve the following for y1, y2 ∈ F = R(x) by viewing it as a system of linear eqns in y1, y2 with co-effs in F. xy1 + y2 = 0 y1 + xy2 = x − 1 Point Gaussian elimination works over any field.

Daniel Chan (UNSW) Lecture 4: Fields Semester 2 2012 10 / 10