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$F_{4} = F_{2} [t] / (t^{2} + t + 1)$ with complete addition and multiplication tables

Example

Let $a$ be the residue class of $t$ in $F_{2} [t] / (t^{2} + t + 1)$ . Then $a^{2} = a + 1$ , and the field has elements $0, 1, a, a + 1$ with tables

$+$	$0$	$1$	$a$	$a + 1$
$0$	$0$	$1$	$a$	$a + 1$
$1$	$1$	$0$	$a + 1$	$a$
$a$	$a$	$a + 1$	$0$	$1$
$a + 1$	$a + 1$	$a$	$1$	$0$

$\cdot$	$0$	$1$	$a$	$a + 1$
$0$	$0$	$0$	$0$	$0$
$1$	$0$	$1$	$a$	$a + 1$
$a$	$0$	$a$	$a + 1$	$1$
$a + 1$	$0$	$a + 1$	$1$	$a$

Facts & Assumptions

Given: The quotient $A = F_{2} [t] / (t^{2} + t + 1)$ and the residue class $a$ of $t$ .

[L1]

For a field $F$ , the quotient $F [t] / (p)$ is a field exactly when the nonconstant polynomial $p$ is irreducible (For a nonconstant $p$ in $F [x]$ , the ideal $(p)$ is maximal and $F [x] / (p)$ is a field exactly when $p$ is irreducible).

[L2]

The ring $Z / 2$ is a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

[L3]

The multiplicative group of a finite field is cyclic (The multiplicative group $F_{q}^{\times}$ of a finite field is cyclic).

Verification

technique · direct

1.1

The polynomial $t^{2} + t + 1$ has value $1$ at both $0$ and $1$ , so it has no root in $F_{2}$ and is irreducible. By [L1] and [L2], $A$ is a field.

givenL1L2

2.1

Every residue has the unique form $c + d a$ with $c, d \in F_{2}$ , and the relation is $a^{2} + a + 1 = 0$ , hence $a^{2} = a + 1$ .

step 1.1algebra

3.1

Applying characteristic-two addition and the reduction in step 2.1 gives every entry in the two displayed tables. In particular $a (a + 1) = 1$ and $(a + 1)^{2} = a$ , so the three nonzero elements form the cyclic group $1, a, a + 1$ predicted by [L3].

step 2.1L3algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 73 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

K. Conrad, Finite Fields, Examples 1.2-1.4 (standard reference, not scraped)

F4=F2[t]/(t2+t+1)F4​=F2​[t]/(t2+t+1) with complete addition and multiplication tables