That's why I avoid modules. Proof of Fermat's theorem doesn't need them, only vector spaces.

Fermat's theorem is lacking a multiplicative product. STR establishes the multiplicative product in each of the two bases for independent variables, and shows that the Binomial Expansion includes multiplicative products between independent variables, and so is complete.

So abstract modules are completely irrelevant to proving Fermat's theorem; they are totally unnecessary if one admits functions according to the normal rules of arithmetic. Arithmetic alone cannot have multiple dimensions, and in the context of Fermat's theorem, such a reduction is a reduction ad absurdum.

Just the declaration of the product ct is an example of multiplication not included in Fermat's expression.

Modules in this sense are a solution looking for a problem...they are not even thought widgets... and even to create a thought widget, there has to be something there a rate of widget creation (call it c). And to declare an invariant one has to terminate the rate of widget creation by a scalar, t, so x = ct

Modules are an attempt to characterize t without c as a rate of change; if c= 0, there is nothing except a tautology.

A straight line has the equation y=Ax + b. For b=0, y=Ax. Even if A = 1, to declare two variables y=x, one needs two dimensions (x,y) unless the expression is a tautology. .

Creating a basis in a single dimension for all possible scalar multiples is accomplished by

(the foundation of the so-called "time dilation" equation, except it is a relation between an initial multiplicative state (ct) a change of state (vt') and a final state (ct) of widget creation, thought process or not.

Dividing both sides by the final state (ct') yields the relativistic unit circle, thus defining sin and cos (if one multiplies by or a relation between squares geometrically, but abstractly it means that is independent of y = x, and the same is true of higher powers, which is why a basis of powers can form a linear system of polynomials with constant coefficients.

To establish a second independent widget, a second circle is needed, where the unit basis is provided in the same manner (with independent field variables c,v,t, and t')

The Binomial Expansion is then a function which can then be compared with Fermat's Expression , the latter of which has no multiplicative terms, and is thus the basis of a Presburger arithmetic.

(Note that the constructivist approach does not include powers; which in one dimension is a tautology as compared with where A is a scaling factor (a slope) even if A = 1.

Modules are just a complicated way of trying to ignore scalars by trying to provide a justification for powers in terms of counting (Peano's axions, group theory independent of dimension - distinguishable variables over two distinct fields).

Modules are an attempt to characterize x=ct (or y=ct) without declaring a rate of change c (so even derivatives are irrelevant). again, c=0 means there is nothing there, not even a thought widget just a tautology as an expression .

Regarding polynomials, independent variables, and Fermat's theorem vs. the Binomial Expansion, the concept of modules without dimension is an abstraction full of sound and fury and signifying nothing (except tautologies).

Correction: vector spaces can be null or one-dimensional (I can't find where I stated the contrary; on the other hand I have used one-dimensional and null vectors throughout this thread). Doesn't change my argument here.

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