ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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$(\begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array})$ over $F_{2}$ has characteristic polynomial $x^{2} + x + 1$ and no eigenvalue in its base field

Example

Over $F_{2}$ , the matrix

$A = (\begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array})$

has characteristic polynomial $x^{2} + x + 1$ and no eigenvalue in $F_{2}$ .

Facts & Assumptions

Given: The displayed matrix over $F_{2} = Z / 2 Z$ .

[L1]

Since $2$ is prime, $Z / 2 Z$ is a field, whose only elements are $0$ and $1$ (For every prime $p$ , the two operations on $Z / p$ make it a field).

[L2]

The spectrum over the base field is exactly the root set in that field of the characteristic polynomial (For every finite-dimensional space, $σ_{F} (T)$ is exactly the set of roots in $F$ of $χ_{T}$ ).

Verification

technique · exhaustive finite computation

1.1

In $F_{2} [x]$ , $χ_{A} (x) = \det (\begin{array}{cc} x & - 1 \\ - 1 & x - 1 \end{array}) = x (x - 1) - 1 = x^{2} + x + 1$ .

L1algebra

2.1

By [L1], the only candidate roots are $0$ and $1$ , and $χ_{A} (0) = 1$ while $χ_{A} (1) = 1 + 1 + 1 = 1$ in $F_{2}$ .

step 1.1L1algebra

3.1

Thus $χ_{A}$ has no root in its base field, and [L2] gives $σ_{F_{2}} (A) = \emptyset$ .

step 2.1L2∎

Depends on

Used by

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(0111)(01​11​) over F2F2​ has characteristic polynomial x2+x+1x2+x+1 and no eigenvalue in its base field