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In characteristic $2$ , $x^2+1=(x+1)^2$ has zero derivative and a repeated root

Example

In $(\mathbb Z/2)[x]$ , the polynomial $f=x^2+1$ has repeated root $1$ and has formal derivative $0$ .

Facts & Assumptions

Given: The quotient field $\mathbb Z/2$ and the polynomial $f=x^2+1$ .

[L1]

A root is repeated exactly when it is also a root of the formal derivative (A root is repeated exactly when it is also a root of the formal derivative).

[L2]

The power and Leibniz rules hold for formal derivatives over any commutative ring (Linearity, power rule, Leibniz rule and the degree bound for the formal derivative).

[L3]

The ring $\mathbb Z/2$ is the quotient ring $\mathbb Z/2\mathbb Z$ , so $2=0$ and $1+1=0$ (For every $n\in\mathbb N$ , the congruence-class ring $\mathbb Z/n$ is the quotient ring $\mathbb Z/n\mathbb Z$ ).

[L4]

An integer is prime when it exceeds $1$ and has no positive divisors other than $1$ and itself (Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$ ).

[L5]

For prime $p$ , the ring $\mathbb Z/p$ is a field (For every prime $p$ , the two operations on $\mathbb{Z}/p$ make it a field).

Verification

technique · direct

1.1

A direct divisor check using [L4] shows that $2$ is prime, so [L5] licenses the field $\mathbb Z/2$ . By [L3], $-1=1$ and $(x-1)^2=(x+1)^2=x^2+2x+1=x^2+1=f$ , so $1$ is a repeated root.

givenL3L4L5algebra

2.1

By [L2], $f'=2x=0$ in $\mathbb Z/2[x]$ , so both $f(1)$ and $f'(1)$ vanish, agreeing with the criterion [L1].

step 1.1L1L2L3∎

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: 90 results over 21 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

Brian Conrad, Differential Criterion and Primitivity, Section 1 (standard reference, not scraped)

In characteristic 222, x2+1=(x+1)2x^2+1=(x+1)^2x2+1=(x+1)2 has zero derivative and a repeated root