Theorem.  The two sets,

 

equation001.

And

equation003.

Since the ring of polynomials is a UFD, we can factor each factor into where

 

equation005.

 

 

equation007.

 

 

Proof.

equation009.

 

A column reduction by Gaussian elimination using mod p arithmetic will reduce Q-I to column-echelon form.

By the a theorem of linear algebra,

The nullity of the left nullspace = n.

 


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