Is the Polynomial X p n − X the zero polynomial in characteristic p?Suppose that p is a prime number and that F p n [ X ] is the ring of polynomials with one variable, with coefficients in the field F p n for some n ∈ N . Then p ( X ) := X p n − X ∈ F p n [ X ]is a polynomial, which appears at many places in number theory, for example in the definition of the Carlitz Exponential. (In books about function field arithmetics this polynomial is often called [1], where in general [ k ] := X p n ⋅ k − X)Now the question is: Since p ( X ) = 0 for any X ∈ F p n , isn't f just the zero polynomial? The same should be true for all [k], but as this is not mentioned in any book on function field arithmetic, I think I might overlook something.Maybe someone has something enlightening to say here.

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Is the Polynomial       X                  p        n              −  X the zero polynomial in characteristic p?Suppose that p is a prime number and that             F                      p        n              [  X  ] is the ring of polynomials with one variable, with coefficients in the field             F                      p        n             for some   n  ∈      N  . Then  p  (  X  )  :=      X                  p        n              −  X  ∈            F                      p        n              [  X  ]is a polynomial, which appears at many places in number theory, for example in the definition of the Carlitz Exponential. (In books about function field arithmetics this polynomial is often called [1], where in general   [  k  ]  :=      X                  p                  n          ⋅          k                      −  X)Now the question is: Since   p  (  X  )  =  0 for any   X  ∈            F                      p        n            , isn&#039;t f just the zero polynomial? The same should be true for all [k], but as this is not mentioned in any book on function field arithmetic, I think I might overlook something.Maybe someone has something enlightening to say here.

cismadmec · Accepted Answer

Step 1No it is not. The zero polynomial has degree   −  ∞ whereas this polynomial has degree       p    n  . Moreover,       x                  p        n              −  x has a unique factorization as it is a polynomial over a field, whereas 0 is infinitely divisible by anything.Step 2The key disconnect here is that you are only considering solutions in             F                      p        n             and you are thinking that a polynomial is determined by its values on too small a set-remember that polynomials of degree n are determined by their values on sets of size   n  +  1, so you have a bunch of zeros, but you are missing a final point to uniquely identify this polynomial, and if you take any field extension of             F                      p        n             you get a non-zero output for that input, which is different from the zero polynomial.

Jenny Schroeder · Accepted Answer

Step 1      X                  p        n              −  X is clearly not the zero polynomial because it has nonzero coefficients.However,       X                  p        n              −  X does induce the zero function.Step 2In fact, the polynomials that induce the zero function are exactly the multiples of       X                  p        n              −  X.In general, if K is a field, then there is a homomorphism of K-algebras   K  [  X  ]  →      K    K   that transforms polynomials into functions. This map is injective iff K is infinite. When K is finite with q elements, the kernel is generated by       X    q    −  X. This is a consequence of Lagrange&#039;s theorem in group theory applied to       K    ×  .

Is the Polynomial X^{p^n}-X the zero polynomial in characteristic p?

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