- 6.1 Algebraic elements
- 6.2 Finite and algebraic extensions
- 6.3 Geometric constructions
- 6.4 Splitting fields
- 6.5 Finite fields
- 6.6 Irreducible polynomials over finite fields
- 6.7 Quadratic reciprocity

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**6.1.2. Definition.**
Let F be an extension field of K and let
u F.
If there exists a nonzero polynomial
f(x) K[x]
such that f(u)=0, then u is said to be
**algebraic**
over K.
If there does not exist such a polynomial, then u is said to be
**transcendental**
over K.

**6.1.3. Proposition.**
Let F be an extension field of K, and let
u F be algebraic over K.
Then there exists a unique monic irreducible polynomial
p(x) K[x]
such that p(u)=0.
It is characterized as the monic polynomial of minimal degree
that has u as a root.

Furthermore, if f(x) is any polynomial in K[x] with f(u)=0, then p(x) | f(x).

**6.1.4. Definition.**
Let F be an extension field of K,
and let u be an algebraic element of F.
The monic polynomial p(x) of minimal degree in K[x] such
that p(u)=0 is called the
**minimal polynomial**
of u over K. The degree of the minimal polynomial of u over K is called the
**degree**
of u over K.

**6.1.5. Definition.**
Let F be an extension field of K, and let
u_{1},
u_{2},
. . . ,
u_{n} F.
The smallest subfield of F that contains K and
u_{1},
u_{2},
. . . ,
u_{n}
will be denoted by

K ( u_{1},
u_{2},
. . . , u_{n} ).

If F = K(u) for a single element uF, then F is said to be a

**6.1.6. Proposition.**
Let F be an extension field of K, and let
u F.

**(a)**If u is algebraic over K, then K(u) K[x]/<p(x)>, where p(x) is the minimal polynomial of u over K.**(b)**If u is transcendental over K, then K(u) K(x), where K(x) is the quotient field of the integral domain K[x].

**6.1.7. Proposition.**
Let K be a field and let
p(x) K[x]
be any irreducible polynomial.
Then there exists an extension field F of K and an element
u F
such that the minimal polynomial of u over K is p(x).

**6.2.2. Definition.**
Let F be an extension field of K.
If the dimension of F as a vector space over K is finite,
then F is said to be a
** finite**
extension of K.

The dimension of F as a vector space over K is called the
** degree**
of F over K, and is denoted by [F:K].

**6.2.3. Proposition.**
Let F be an extension field of K and let
u F.
The following conditions are equivalent:

**(1)**u is algebraic over K;**(2)**K(u) is a finite extension of K;**(3)**u belongs to a finite extension of K.

[F:K] = [F:E][E:K].

**6.2.6 Corollary.**
Let F be an extension field of K, with algebraic elements
u_{1},
u_{2},
. . . ,
u_{n} F.
Then the degree of

K ( u_{1},
u_{2},
. . . ,
u_{n} )

**6.2.7. Corollary.**
Let F be an extension field of K.
The set of all elements of F that are algebraic over K forms a subfield of F.

**6.2.8. Definition.**
An extension field F of K is said to be
** algebraic**
over K if each element of F is algebraic over K.

**6.2.9. Proposition.**
Every finite extension is an algebraic extension.

**Example.** 6.2.3. (Algebraic numbers)
Let **Q*** be the set of all complex numbers
u **C**
such that u is algebraic over **Q**.
Then **Q*** is a subfield
of **C** by Corollary 6.2.7, called the
**field of algebraic numbers**.

**6.2.10. Proposition.**
Let F be an algebraic extension of E and
let E be an algebraic extension of K.
Then F is an algebraic extension of K.

**6.3.2. Proposition.**
The set of all constructible real numbers
is a subfield of the field of all real numbers.

**6.3.3. Definition.**
Let F be a subfield of **R**.
The set of all points (x,y) in the Euclidean plane
**R**^{2}
such that x,y
F is called the
**plane**
of F.

A straight line with an equation of the form
ax+by+c = 0, for elements a,b,c F,
is called a
**line in F**.

Any circle with an equation of the form
x^{2} + y^{2} + ax + by + c = 0, for
elements a,b,c F,
is called a
**circle in F**.

**6.3.4. Lemma.**
Let F be a subfield of **R**.

**(a)**Any straight line joining two points in the plane of F is a line in F.**(b)**Any circle with its radius in F and its center in the plane of F is a circle in F.

**6.3.6. Theorem.**
The real number u is constructible if and only if
there exists a finite set
u_{1},
u_{2},
. . . ,
u_{n}
of real numbers such that

**(i)**u_{1}^{2}**Q**,**(ii)**u_{i}^{2}**Q**(u_{1},...,u_{i-1}), for i=2,..., and**(iii)**u**Q**(u_{1},...,u_{n}).

**6.3.9. Theorem.**
It is impossible to find a general construction for trisecting an angle,
duplicating a cube, or squaring a circle.

**(i)**f(x) = a_{n}(x-r_{1}) (x-r_{2})**· · ·**(x-r_{n}), and**(ii)**F = K(r_{1},r_{2},...,r_{n}).

**6.4.2. Theorem.**
Let
f(x) K[x] be a polynomial of degree
n>0.
Then there exists a splitting field F for f(x) over K, with
[F:K] n!.

**6.4.3. Lemma.**
Let
: K -> L be an isomorphism of fields.
Let F be an extension field of K such
that F = K(u) for an algebraic element
u F.
Let p(x) be the minimal polynomial of u over K.
If v is any root of the image q(x) of p(x) under
,
and E=L(v), then there is a unique way to extend
to an isomorphism
: F -> E such that
(u) = v and
(a) =
(a)
for all a K.

**6.4.5. Theorem.**
Let f(x) be a polynomial over the field K.
The splitting field of f(x) over K is unique up to isomorphism.

If F is any field, then the smallest subfield of F that
contains the identity element 1 is called the
**prime subfield**
of F.
If F is a finite field, then its prime subfield is isomorphic to
**Z**_{p},
where p=chr(F) for some prime p.

**6.5.2. Theorem.**
Let F be a finite field with
k = p^{n}
elements. Then F is the splitting field of the polynomial
x^{k}-x over the prime subfield of F.

**Example** 6.5.1. **[Wilson's theorem]**
Let p > 2 be a prime number. Then

(p-1)! -1 (mod p).

**6.5.4. Lemma.**
Let F be a field of prime characteristic p, let
n **Z**^{+},
and let k = p^{n}. Then

{ a
F | a^{k} = a }

**6.5.5. Proposition.**
Let F be a field with
p^{n}
elements. Each subfield of F has
p^{m}
elements for some divisor m of n.
Conversely, for each positive divisor m of n there
exists a unique subfield of F with
p^{m}
elements.

**6.5.6. Lemma.**
Let F be a field of characteristic p.
If n is a positive integer not divisible by p, then the polynomial
x^{n}-1
has no repeated roots in any extension field of F.

**6.5.7. Theorem.**
For each prime p and each positive integer n, there exists a field with
p^{n} elements.

**6.5.8. Definition.**
Let p be a prime number and let
n **Z**^{+}.
The field (unique up to isomorphism) with
p^{n}
elements is called the
**Galois field of order p**^{n},
denoted by GF(p^{n}).

**6.5.9. Lemma.**
Let G be a finite abelian group. If
a G is an element of maximal order in G,
then the order of every element of G is a divisor of the order of a.

**6.5.10. Theorem.**
The multiplicative group of nonzero elements of a finite field is cyclic.

**6.5.11. Theorem.**
Any finite field is a simple extension of its prime subfield.

**6.5.12. Corollary.**
For each positive integer n there exists an
irreducible polynomial of degree n over GF(p).

**Convention:**
In the notation
_{d | n}
and
_{d | n}
we will assume that d | n refers to the positive divisors of n.

**6.6.2. Definition.**
If d is a positive integer, we define the
**Moebius**
function
(d) as follows:

- (1) = 1;
- (d) = 1
if d has an even number of prime factors (each occurring only once);
- (d) = -1
if d has an odd number of prime factors (each occurring only once);
- (d) = 0 if d is divisible by the square of a prime.

(mn) = (m) (n).

If R is a commutative ring, then a function f :f(mn) = f(m)f(n),

whenever gcd(m,n)=1.
**6.6.4. Proposition.**
Let R be a commutative ring, and let
f : **Z**^{+} -> R
be a multiplicative function. If
F : **Z**^{+} -> R
is defined by

F(n) =
_{d | n}
f(d),

**6.6.5. Proposition.**
For any positive integer n,

_{d | n}
(d) = 1 if n = 1, and

_{d | n}
(d) = 0 if n > 1.

F(n) =
_{d | n}
f(d),
for all
n **Z**^{+},

f(m)
=
_{n | m}
( m/n ) F(n),
for all
m **Z**^{+}.

G(n) =
_{d | n}
g(d), for all
n **Z**^{+},

g(m) =
_{n | m}
G(n)^{k},
for all
m **Z**^{+},

**6.6.8. Definition.**
The number of irreducible polynomials of degree m
over the finite field GF(q), where q is a prime power,
will be denoted by I_{q}(m).

The following formula for I_{q}(m) is due to Gauss.

**6.6.9. Theorem.**
For any prime power q and any positive integer m,

I_{q}(m) = (1/m)
_{d | m}
( m / d ) q^{d}.

When n is a prime, we write =1 if a is a quadratic residue modulo n and =-1 if a is a quadratic nonresidue modulo n. The symbol is called the

**6.7.2. Proposition. [Euler's Criterion]**
If p is an odd prime, and if
a **Z** with
p a, then

**(i)**, and**(ii)**, where k = (p^{2}- 1)/8.