See: Field with one element
Understand finite fields and what {$F_{1^n}$} could mean.
Understand the structure of finite fields
- Relate them to Lyndon words and symmetric functions of eigenvalues.
- Consider what "infinity" Q means in a finite field. Then consider what happens to Q as we take q to infinity.
- In the equation for the Riemann formula, where it allows for powers of prime and weights them, is it related to finite fields?
- Compare finite field behavior (division winding around) with complex number behavior (winding around).
- Sieja nepažymėtą priešingybę (+) -1, 0, 1 ir pažymėtą priešingybę (x) -1, 1.
- W: Field with one element
- The nonexistent element of {$F_1$} may be considered to not exist, or imagined to exist, but regardless, I expect that cognitively there are three ways to interpret it as 0, 1, ∞, which thereby expand upon the duality between existence and nonexistence and make it structurally richer.
- Relate elliptic transforms to God's dance {0, 1, ∞} and {$F_1$}: There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: {$1 / z$}, which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), {$1 − z$} which fixes ∞ and swaps 0 with 1 (rotation by 180° about the points 1/2 and ∞), and {$z / ( z − 1 )$} which fixes 0 and swaps 1 with ∞ (rotation by 180° about the points 0 and 2). Note that this relates pairs from: 1, z, z-1.
- Study how turning the counting around relates to cycles - finite fields.