ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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In characteristic $2$ , $x^{2} + 1 = (x + 1)^{2}$ has zero derivative and a repeated root

Example

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

Facts & Assumptions

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

[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 $Z / p$ is a field (For every prime $p$ , the two operations on $Z / p$ 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 $Z / 2$ . By [L3], $- 1 = 1$ and $(x - 1)^{2} = (x + 1)^{2} = x^{2} + 2 x + 1 = x^{2} + 1 = f$ , so $1$ is a repeated root.

givenL3L4L5algebra

2.1

By [L2], $f^{'} = 2 x = 0$ in $Z / 2 [x]$ , 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 · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

In characteristic 22, x2+1=(x+1)2x2+1=(x+1)2 has zero derivative and a repeated root