Field With One Element - Field Extensions

Field Extensions

One may define field extensions of the field with one element as the group of roots of unity, or more finely (with a geometric structure) as the group scheme of roots of unity. This is non-naturally isomorphic to the cyclic group of order n, the isomorphism depending on choice of a primitive root of unity:

Thus a vector space of dimension d over F_1n is a finite set of order dn on which the roots of unity act freely, together with a base point.

From this point of view the finite field F_q is an algebra over F_1n, of dimension d = (q − 1)/n for any n that is a factor of q − 1 (for example n = q − 1 or n = 1). This corresponds to the fact that the group of units of a finite field F_q (which are the q − 1 non-zero elements) is a cyclic group of order q − 1, on which any cyclic group of order dividing q − 1 acts freely (by raising to a power), and the zero element of the field is the base point.

Similarly, the real numbers R are an algebra over F₁₂, of infinite dimension, as the real numbers contain ±1, but no other roots of unity, and the complex numbers C are an algebra over F_1n for all n, again of infinite dimension, as the complex numbers have all roots of unity.

From this point of view, any phenomenon that only depends on a field having roots of unity can be seen as coming from F₁ – for example, the discrete Fourier transform (complex-valued) and the related number-theoretic transform (Z/nZ-valued).