Twisted polynomial ring

In mathematics, a twisted polynomial{{Cite book |last=Saltman |first=David J. |url=https://books.google.com/books?id=2sIShCIZ_uUC&dq=Twisted+polynomial+ring&pg=PA52 |title=Lectures on Division Algebras |publisher=American Mathematical Soc. |isbn=978-0-8218-8938-1 |language=en}}{{Cite book |last1=Gardner |first1=J. W. |url=https://books.google.com/books?id=opoxhIF-3moC&dq=Twisted+polynomial+ring&pg=PA115 |title=Radical Theory of Rings |last2=Wiegandt |first2=R. |date=2003-11-19 |publisher=CRC Press |isbn=978-0-203-91335-2 |language=en}} is a polynomial over a field of characteristic $p$ in the variable $\tau$ representing the Frobenius map $x\mapsto x^p$ . In contrast to normal polynomials, multiplication of these polynomials is not commutative, but satisfies the commutation rule

: $\tau x=x^p \tau$

for all $x$ in the base field.

Over an infinite field, the twisted polynomial ring is isomorphic to the ring of additive polynomials, but where multiplication on the latter is given by composition rather than usual multiplication. However, it is often easier to compute in the twisted polynomial ring — this can be applied especially in the theory of Drinfeld modules.

Definition

Let $k$ be a field of characteristic $p$ . The twisted polynomial ring $k\{\tau\}$ is defined as the set of polynomials in the variable $\tau$ and coefficients in $k$ . It is endowed with a ring structure with the usual addition, but with a non-commutative multiplication that can be summarized with the relation $\tau x=x^p\tau$ for $x\in k$ . Repeated application of this relation yields a formula for the multiplication of any two twisted polynomials.{{Cite book |last1=Gardner |first1=J. W. |url=https://books.google.com/books?id=opoxhIF-3moC&dq=Twisted+polynomial+ring%2C+definition&pg=PA18 |title=Radical Theory of Rings |last2=Wiegandt |first2=R. |date=2003-11-19 |publisher=CRC Press |isbn=978-0-203-91335-2 |language=en}}

As an example we perform such a multiplication

: $(a+b\tau)(c+d\tau)=a(c+d\tau)+b\tau(c+d\tau)=ac+ad\tau+bc^p\tau+bd^p\tau^2$

Properties

The morphism

: $k\{\tau\}\to k[x],\quad a_0+a_1\tau+\cdots+a_n\tau^n\mapsto a_0x+a_1x^p+\cdots+a_nx^{p^n}$

defines a ring homomorphism sending a twisted polynomial to an additive polynomial. Here, multiplication on the right hand side is given by composition of polynomials.{{Cite book |last=Mora |first=Teo |url=https://books.google.com/books?id=8DjzCwAAQBAJ&dq=Twisted+polynomial+ring%2C+Properties&pg=PA257 |title=Solving Polynomial Equation Systems IV: Volume 4, Buchberger Theory and Beyond |date=2016-04-01 |publisher=Cambridge University Press |isbn=978-1-316-38138-0 |language=en}} For example

: $(ax+bx^p)\circ (cx+dx^p)=a(cx+dx^p)+b(cx+dx^p)^p=acx+adx^p+bc^px^p+bd^px^{p^2},$

using the fact that in characteristic $p$ we have the Freshman's dream $(x+y)^p=x^p+y^p$ .

The homomorphism is clearly injective, but is surjective if and only if $k$ is infinite. The failure of surjectivity when $k$ is finite is due to the existence of non-zero polynomials which induce the zero function on $k$ (e.g. $x^q-x$ over the finite field with $q$ elements).{{fact|reason=The relations deriving polynomial equality from pointwise equality have not been assumed.|date=March 2015}}

Even though this ring is not commutative, it still possesses (left and right) division algorithms.

References

{{Citation | last1=Goss | first1=D. | authorlink = David Goss | title=Basic structures of function field arithmetic | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] | isbn=978-3-540-61087-8 | mr=1423131 | year=1996 | volume=35 | zbl=0874.11004 }}
{{citation | title=Number Theory in Function Fields | volume=210 | series=Graduate Texts in Mathematics | issn=0072-5285 | first=Michael | last=Rosen | publisher=Springer-Verlag | year=2002 | isbn=0-387-95335-3 | zbl=1043.11079 }}

Category:Algebraic number theory

Category:Finite fields