Field with 4 Elements has only Order 2 Elements
Theorem
Let $\struct {\GF, +, \times}$ be a field which has exactly $4$ elements.
Then:
- $\forall a \in \GF: a + a = 0_\GF$
where $0_\GF$ is the zero of $\GF$.
Proof 1
By definition of field, both the algebraic structures $\struct {\GF, +}$ and $\struct {\GF^*, \times}$ are (abelian) groups, where $\GF^* := \GF \setminus \set 0$.
By definition:
From Classification of Groups of Order up to 15, there are only two possibilities:
- $(1): \quad \struct {\GF, +} \cong \Z_4$ and $\struct {\GF^*, \times} \cong \Z_3$
- $(2): \quad \struct {\GF, +} \cong \Z_2 \times \Z_2$ and $\struct {\GF^*, \times} \cong \Z_3$.
In case $(2)$:
- $\forall a \in \GF: a + a = 0_\GF$
but in case $(1)$:
- $\exists a, b \in \GF: a + a = b \ne 0_\GF$ where $b + b = 0_\GF$
as there exists an element of $\struct {\GF, +}$ of order $4$.
Aiming for a contradiction, suppose $(1)$ describes a field.
Then:
\(\ds \paren {b + b} \times b\) | \(=\) | \(\ds 0_\GF \times b\) | as $b + b = 0_\GF$ from above | |||||||||||
\(\ds \) | \(=\) | \(\ds 0_\GF\) | as $0_\GF$ is the zero of $\struct {\GF, +, \times}$ |
But then:
\(\ds \paren {b + b} \times b\) | \(=\) | \(\ds \paren {b \times b} + \paren {b \times b}\) | as $\times$ is distributive over $+$ |
As $\struct {\GF, +} \cong \Z_4$, both $a + a = b$ and $\paren {-a} + \paren {-a} = b$.
The only way for $\paren {b \times b} + \paren {b \times b} = 0_\GF$ is for $b \times b = b$ or $b \times b = 0_\GF$.
The second case is eliminated as $0_\GF \notin \struct {\GF^*, \times}$.
So it must be the case that $b \times b = b$.
So as $\struct {\GF^*, \times} \cong \Z_3$ it must follow that $a \times b = a, a \times \paren {-a} = b$ and then $a \times a = -a$.
It follows that the Cayley tables of $\struct {\GF, +}$ and $\struct {\GF^*, \times}$ must therefore be as follows:
- $\begin{array}{c|cccc}
+ & 0_\GF & a & b & -a \\ \hline 0_\GF & 0_\GF & a & b & -a \\ a & a & b & -a & 0_\GF \\ b & b & -a & 0_\GF & a \\ -a & -a & 0_\GF & a & b \\ \end{array} \qquad \begin{array}{c|cccc} \times & 0_\GF & b & a & -a \\ \hline 0_\GF & 0_\GF & 0_\GF & 0_\GF & 0_\GF \\ b & 0_\GF & b & a & -a \\ a & 0_\GF & a & -a & b \\ -a & 0_\GF & -a & b & a \\ \end{array}$
\(\ds \paren {a + b} \times a\) | \(=\) | \(\ds \paren {-a} \times a\) | as $a + b = -a$ from above | |||||||||||
\(\ds \) | \(=\) | \(\ds b\) | as $\paren {-a} \times a = b$ from above |
But:
\(\ds \paren {a \times a} + \paren {b \times a}\) | \(=\) | \(\ds \paren {-a} + a\) | from above | |||||||||||
\(\ds \) | \(=\) | \(\ds 0_\GF\) | from above |
So:
- $\paren {a + b} \times a \ne \paren {a \times a} + \paren {b \times a}$
demonstrating that $\times$ is not distributive over $+$.
Thus $\GF$ as has been defined:
- $\struct {\GF, +} \cong \Z_4$ and $\struct {\GF^*, \times} \cong \Z_3$
is not a field.
It follows that our supposition that $\struct {\GF, +} \cong \Z_4$ was false.
Thus, if $\GF$ is a field, then $\struct {\GF, +} \cong \Z_2 \times \Z_2$.
That is:
- $\forall a \in \GF: a + a = 0_\GF$
as needed to be proved.
$\blacksquare$
Proof 2
Let $\struct {\GF, +, \times}$ be a field of order $4$ whose zero is $0_\GF$.
By definition, $\GF$ is a Galois field.
The additive group $\struct {\GF, +}$ of $\GF$ can be one of two:
- $(1): \quad$ the cyclic group of order $4$, generated by the identity of $\struct {\GF, +}$ which is $0_\GF$
or:
- $(2): \quad$ the Klein $4$-group, whose elements are all of the form $a + a = 0_\GF$.
Aiming for a contradiction, suppose $(1)$ is the additive group of a field.
Then the characteristic of $\GF$ is $4$.
But from Characteristic of Galois Field is Prime it follows that $\GF$ is not a field.
From that contradiction it follows that the additive group $\struct {\GF, +}$ of $F$ cannot be the cyclic group of order $4$.
Hence for $\GF$ to be a field at all, it is necessary for $\struct {\GF, +}$ to be the Klein $4$-group.
$\blacksquare$
Also see
- Galois Field of Order $4$, where it is demonstrated that such an $\GF$:
- $\struct {\GF, +} \cong \Z_2 \times \Z_2$ and $\struct {\GF^*, \times} \cong \Z_3$
- is actually a field.
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Exercise $5$