ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-16

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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$F_{9} = F_{3} [t] / (t^{2} + 1)$ and a generator of its multiplicative group

Example

In $F_{9} = F_{3} [t] / (t^{2} + 1)$ , write $a$ for the class of $t$ . Then $a^{2} = 2$ and $a$ has order $4$ , while $1 + a$ has order $8$ and generates $F_{9}^{\times}$ .

Facts & Assumptions

Given: The quotient $A = F_{3} [t] / (t^{2} + 1)$ and the class $a$ of $t$ .

[L1]

A polynomial quotient over a field is a field exactly when its modulus 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 $Z / 3$ is a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

[L3]

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

Verification

technique · direct

1.1givenL1L2

The squares in $F_{3}$ are $0$ and $1$ , so $t^{2} + 1$ has no root and is irreducible. Thus [L1] and [L2] make $A$ a field with the nine elements $r + s a$ , where $r, s \in F_{3}$ , and $a^{2} = 2$ .

2.1step 1.1algebra

One has $a^{2} = 2$ and $a^{4} = 1$ , with $a^{2} \neq 1$ , so $a$ has order $4$ . Put $b = 1 + a$ . Then $b^{2} = 2 a$ , $b^{4} = 2$ , and $b^{8} = 1$ .

3.1step 2.1L3∎

Since $b^{2} \neq 1$ and $b^{4} \neq 1$ , its order is $8$ . Hence it exhausts the eight nonzero elements and is a generator, as [L3] guarantees some element must be.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

K. Conrad, Finite Fields, Section 1 (standard reference, not scraped)

F9=F3[t]/(t2+1)F9​=F3​[t]/(t2+1) and a generator of its multiplicative group