In this post we will understand field extensions using examples. We will start with basic definitions and develop our understanding of the topic. Let us start with Groups.

Groups#

A group is a set of elements $G = \{a, b, c,...\}$ and an operation $\oplus$ for which the following axioms hold:

• Closure: for any $a, b \in G$, the element $a \oplus b$ is in $G$
• Associative: for any $a, b, c \in G$, $(a \oplus b) \oplus c = a \oplus (b \oplus c)$
• Identity: There is an identity element $e$ in $G$ for which $a \oplus e = e \oplus a = a$, for all $a \in G$
• Inverse: For each $a \in G$, there is an inverse $(-a)$ such that $a \oplus (-a) = e$

A group $G$ for which $a \oplus b = b \oplus a$, for all $a, b \in G$ is called abelian or commutative.

Some examples of groups:

1. Set of integers $\mathbb{Z}$ with addition operation $+$ forms an abelian group.
2. Set $\{0,1,2,....,n-1\}$ denoted by $\mathbb{Z}_n$ under mod-$n$ addition, where $n$ is a positive integer

We will now introduce Fields.

Fields#

A field is a set $\mathbb{F}$ of at least two elements, with two binary operation $+$ and $*$, for which following axioms are satisfied :

• The set $\mathbb{F}$ forms an abelian group (whose identity is called 0) under the operation $+$
• The set $\mathbb{F}^* = \mathbb{F} - \{0\} = \{a \in \mathbb{F}, a \neq 0\}$ forms an abelian group (whose identity is called 1) under the operation $*$
• Distributivity: for all $x, y, z \in \mathbb{F}$, $x*(y+z) = x*y +x*z$

Some examples of fields:

1. Set of real numbers $\mathbb{R}$ under addition and multiplication
2. Set of complex numbers $\mathbb{C}$ under addition and multiplication
3. Set of rational numbers $\mathbb{Q}$ under addition and multiplication
4. Set $\{0,1\}$ under mod-$2$ addition and multiplication
5. Set $\{0,1,2,....,p-1\}$ denoted by $\mathbb{F}_p$ under mod-$p$ addition and multiplication, where $p$ is a prime number. These are called Prime Fields.

We can verify that all the above examples satisfy the field axioms. The sets in example (4) and (5) are finite thus they are examples of a finite field. Now let’s state a theorem on extension of Prime Fields.

Theorem 1: Let $\mathbb{N}$ denote set of natural numbers. For all primes $p \in \mathbb{N}$ and $k \in \mathbb{N}, k \geq 1$ there exists a field with $p^k$ elements denoted by $\mathbb{F}_{p^k}$. $\mathbb{F}_{p^k}$ is called degree $k$ extension of $\mathbb{F}_p$.

We will exclude the proof of above theorem since the main objective of this post is to understand field extensions.

Let us go through few non-examples of fields :

1. Set of integers $\mathbb{Z}$ under addition and multiplication is not an example of field since multiplicative inverse does not exist for all non-zero elements of $\mathbb{Z}$.
2. Let $p=2$ and $k=2$ in Theorem 1. The set $\mathbb{F}_4 = \{0,1,2,3\}$ under mod-$4$ addition and multiplication is not a field since there is no multiplicative inverse for element $2$ i.e there does not exist an element $x\in \mathbb{F}_4$ such that $2*x$ mod $4 = 1$

In the above example, we saw that $\mathbb{F}_4 = \{0,1,2,3\}$ is not a field but as per Theorem 1 there exists a field with $p=2$ and $k=2$. We will see the construction of this field in further sections and in general see the construction of $\mathbb{F}_{p^k}$. But before that let’s introduce Commutative Rings.

Commutative Rings#

A commutative ring $R$ is a set with two binary operations $+$ and $*$, which satisfies the following axioms:

• The set $R$ forms an abelian group (whose identity is called 0) under the operation $+$
• The set $R$ is closed under multiplication i.e. for any $a, b \in R$, the element $a * b$ is in $R$
• Multiplicative Identity: for all $a \in R$, there is an multiplicative identity $1$ in $R$ for which $a * 1 = 1 * a = a$
• Multiplication is associative i.e. for all $a, b, c \in R$, $a*(b*c) = (a*b)*c$
• Multiplication is commutative i.e. for all $a, b\in R$, $a*b = b*a$
• Distributivity: for all $a, b, c \in R$, $a*(b+c) = a*b +a*c$

We can see that Commutative Rings satisfy all the Field axioms excluding the multiplicative inverse property i.e. in a commutative ring all non-zero elements need not have a multiplicative inverse.

Some examples of commutative ring :

1. Set of integers $\mathbb{Z}$ under addition and multiplication
2. Set of polynomials in variable $X$ with integer coefficients denoted by $\mathbb{Z}[X]$, under polynomial addition and multiplication
3. Set of polynomials in variable $X$ with coefficients in field $\mathbb{F}$ denoted by $\mathbb{F}[X]$, under polynomial addition and multiplication

We can verify that the above examples satisfy all the field axioms except that there are elements in these sets which do not have a multiplicative inverse.

Now before we construct $\mathbb{F}_{p^k}$, the degree $k$ extension of $\mathbb{F}_p$, let us go through polynomials.

Polynomials#

Let us start by defining the degree of a polynomial. Let $P$ be a polynomial such that

then the degree of $P$ denoted by $deg(P)$ is defined as $deg(P) = \underset{i}{\mathrm{max}} \{f_i \neq 0\}$.

Let $R$ be a commutative ring. Then $R[X]$ denotes the ring of polynomials in variable $X$ with coefficients in $R$.

A polynomial of degree $\leq d$ in $R[X]$ is given by

where $f_0, f_1, f_2,....,f_d \in R$ and $f_d \neq 0$.

$R[X]$ is the union of all the polynomials over all degrees in $\mathbb{N} \bigcup 0$.

Next, we will go through two important results regarding polynomials.

Lemma 1: Let $\mathbb{F}$ be a field. Let $\mathbb{F}[X]$ denote the ring of polynomials in variable $X$ with coefficients in $\mathbb{F}$. For all $g$, $h$ $\in \mathbb{F}[X]$ where $h$ is non-zero there exists unique $q$, $r$ $\in \mathbb{F}[X]$ such that $g = h \cdot q + r \;\;\text{and}\;\; deg(r) < deg(h)$.

Lemma 2: Let $g$, $h$ $\in \mathbb{F}[X]$ where $g$ and $h$ are non-zero. Let $I$ be a set such that $I(g,h) = \{u \cdot g + v \cdot h \mid u,v \in \mathbb{F}[X] \}$. Let $p$ be a non-zero polynomial of lowest degree in $I(g,h)$. Then $p$ is a $GCD(g,h)$.

Note that Lemma 1 is similar to Euclid’s division lemma for integers and Lemma 2 is similar to extended euclidean algorithm for integers. Now we will define irreducible polynomials which are used in the definition of extension field $\mathbb{F}_{p^k}$.

Irreducible Polynomials: Let $f(X) \in \mathbb{F}[X]$ be a polynomial with coefficients in $\mathbb{F}$. $f(X)$ is irreducible over $\mathbb{F}$ if $f(X)$ cannot be written as a product of two polynomials $g(X)*h(X)$ where $g,h \in \mathbb{F}[X]$ and $deg(g), \; deg(f) \geq 1$.

Examples:

1. Let $\mathbb{F}$ be a field of real numbers $\mathbb{R}$ then the polynomial $f(X) = X^2 + 3X + 2$ is reducible over $\mathbb{R}$ since $f(X) = X^2 + 3X + 2 = (X+1)*(X+2)$
2. Let $\mathbb{F}$ be a field of real numbers $\mathbb{R}$ then $g(X)=X^2+1$ is irreducible over $\mathbb{R}$ since it cannot be written as a product of two polynomials of degree greater than equal to 1 with coefficients in $\mathbb{R}$.
3. Let $\mathbb{F}$ be a field of complex numbers $\mathbb{C}$ then the polynomial $g(X)=X^2+1$ is reducible over $\mathbb{C}$ since $g(X)=X^2+1 = (X-i)*(X+i)$ where $i=\sqrt{-1}$.

Now we will state an important result about irreducible polynomials.

Lemma 3: For all primes $p$ and $d \in \mathbb{N}$, there exists an irreducible polynomial of degree equal to $d$ in $\mathbb{F}_p[X]$ where $\mathbb{F}_p$ is the field $\{0,1,2,...,p-1\}$ and $\mathbb{F}_p[X]$ denotes the ring of polynomials over $X$ with coefficients in $\mathbb{F}_p$.

Now a polynomial of degree $\leq d$ in $\mathbb{F}_p$ can be written as $f(X) = f_0 + f_1 X + f_2 X^2 + .... + f_d X^d$. The number of polynomials of degree $\leq d$ is $p^{(d+1)}$, since there are total $d+1$ coefficients and each coefficient can take $p$ values from $\mathbb{F}_p$.

Field Extension#

Now we will construct $\mathbb{F}_{p^k}$ using the theorems and definitions we have seen so far.

For $k=1$, $\mathbb{F}_{p}$ is the set $\{0,1,2,3,...,p-1\}$ with mod-$p$ addition and multiplication operations defined on it. These are Prime Fields which we saw in earlier section.

For $k \in \mathbb{N}$ and $k \neq 1$, let $g(X)$ be a degree $k$ irreducible polynomial with coefficients in $\mathbb{F}_{p}$ i.e. $g(X) \in \mathbb{F}_p[X]$ and $deg(g)=k$, from Lemma 3 we know that such a polynomial exists. Then $\mathbb{F}_{p^k}$ is defined as follows :

1. Elements are from the set $\{f(X) \in \mathbb{F}_p[X], deg(f) < k\}$. The number of elements = number of polynomial of $deg(f) < k$ in $\mathbb{F}_p[X]$ = $p^k$

2. Operations are polynomial addition and multiplication modulo $g(X)$

Now let us check that if the above definition of $\mathbb{F}_{p^k}$ satisfies the field axioms.

Commutativity, Associativity and Distributivity of $*$ on $+$ follows directly from the polynomial addition and multiplication modulo $g(X)$ operations. Also, $0$ is the additive inverse and $1$ is the multiplicative inverse.

Existence and Uniqueness of additive inverse: For every $f(X) \in \mathbb{F}_p[X], deg(f) < k$ there exists $h(X) \in \mathbb{F}_p[X], deg(h) < k$ such that $f(X) + h(X) = g(X)$.

Taking modulo $g(X)$ on both sides $(f(X) +h(X)) \;\text{mod-} g(X)=0$ i.e. $h(X)$ is additive inverse of $f(X)$. This follows from the fact that coefficients of $f(X)$ are in $\mathbb{F}_p$, thus they have additive inverse which are the corresponding coefficients of $h(X)$.

Existence and Uniqueness of multiplicative inverse: For every $f(X) \in \mathbb{F}_p[X], deg(f) < k$ there exists $h(X) \in \mathbb{F}_p[X], deg(h) < k$ such that $f(X) * h(X) = 1 \;\text{mod-} g(X)$. Moreover such an $h(X)$ is unique.

$f(X) \in \mathbb{F}_p[X], deg(f) < k$ and $g(X) \in \mathbb{F}_p[X], deg(g) = k$ is an irreducible polynomial thus $GCD(f,g) = 1$. Using Lemma 2,

Thus $\exists u,v \in \mathbb{F}_p[X]$ such that $u(X) \cdot f(X) + v(X) \cdot g(X) = 1$.

Taking mod $g(X)$ on both sides, $u(X) \cdot f(X) = 1 \;\text{mod-} g(X)$. Thus $u(X)$ is an inverse of $f(X)$.

We can further reduce $u(X)$ using Lemma 1, $\exists q,r \in \mathbb{F}_p[X]$ s.t. $u = q \cdot g + r$ and $deg(r) < deg(g)=k$. Thus $(q \cdot g + r) f = 1 \;\text{mod-} g$. Now $r \cdot f = 1 \;\text{mod-} g$. Thus $r$ is the multiplicative inverse of $f$ in $\mathbb{F}_{p^k}$.

We have just shown that $\mathbb{F}_{p^k}$ satisfies the field axioms. $\mathbb{F}_{p^k}$ is called a degree $k$ extension of $\mathbb{F}_{p}$. Note that $\mathbb{F}_{p} \subseteq \mathbb{F}_{p^k}$ i.e. $\mathbb{F}_{p}$ is prime subfield of $\mathbb{F}_{p^k}$. Note that $\forall \alpha \in \mathbb{F}_{p^k}$,

$p$ is called the characteristic of $\mathbb{F}_{p^k}$. By convention characteristic of $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{Q}$ is $0$.

Example: Let $p=2$ and $k=2$ i.e. we will construct $\mathbb{F}_{2^2}=\mathbb{F}_{4}$ using the above definitions.

Let $g(X)$ be a degree $k=2$ irreducible polynomial with coefficients in $\mathbb{F}_{2}$ i.e. $g(X) \in \mathbb{F}_2[X]$ and $deg(g)=k=2$.

Note that there are four polynomial of degree 2 in $\mathbb{F}_2[X]$ and the only irreducible polynomial is $X^2+ X +1$. The other three polynomials are reducible. Since

Thus $g(X)=X^2+ X +1$.

Now total number of elements in $\mathbb{F}_{2^2}$ is $p^k=4$. The elements are from the set $\{f(X) \in \mathbb{F}_2[X], deg(f)< 2\}$ which is precisely the set $\{ 0, 1, X, X+1\}$.

The operations defined on $\mathbb{F}_{2^2}$ are polynomial addition and multiplication modulo $g(X)=X^2+ X +1$.

Let us add two elements from the set.

The last equality follows from the fact that all coefficients are mod-$p$. The addition table for all the field elements is as follows:

Let us multiply two elements from the set.

The last equality follows from polynomial long division. The multiplication table for all the field elements is as follows:

This concludes the post on field extensions. We gradually developed our understanding of the topic starting from definition of Groups to a basic example of field extension.

References#

1. Chapter 7: Introduction to Finite Fields, Course notes by Prof. David Forney
2. Chapter 2: Elliptic Curve Cryptography, An Introduction to Bitcoin by Prof. Saravanan Vijayakumaran