Additive polynomial
Template:One source In mathematics, the additive polynomials are an important topic in classical algebraic number theory.
Definition
Let <math>k</math> be a field of prime characteristic <math>k</math>. A polynomial <math>P(x)</math> with coefficients in <math>k</math> is called an additive polynomial, or a Frobenius polynomial, if
<math display=block>P(a+b)=P(a)+P(b)</math>
as polynomials in <math>a</math> and <math>b</math>. It is equivalent to assume that this equality holds for all <math>a</math> and <math>b</math> in some infinite field containing <math>k</math>, such as its algebraic closure.
Occasionally absolutely additive is used for the condition above, and additive is used for the weaker condition that <math>P(a+b)=P(a)+P(b)</math> for all <math>a</math> and <math>b</math> in the field.<ref name=Goss-p1>Template:Citation</ref> For infinite fields the conditions are equivalent,Template:Sfn but for finite fields they are not, and the weaker condition is the "wrong" as it does not behave well. For example, over a field of order <math>q</math> any multiple <math>P</math> of <math>x^q-x</math> will satisfy <math>P(a+b)=P(a)+P(b)</math> for all <math>a</math> and <math>b</math> in the field, but will usually not be (absolutely) additive.
Examples
The polynomial <math>x^p</math> is additive.<ref name=Goss-p1/> Indeed, for any <math>a</math> and <math>b</math> in the algebraic closure of <math>k</math> one has by the binomial theorem
<math display=block>(a+b)^p = \sum_{n=0}^p {p \choose n} a^n b^{p-n}.</math>
Since <math>p</math> is prime, for all <math>n=1,\dots,p-1</math> the binomial coefficient <math>\tbinom{p}{n}</math> is divisible by <math>p</math>, which implies that
<math display=block>(a+b)^p \equiv a^p+b^p \mod p</math>
as polynomials in <math>a</math> and <math>b</math>.<ref name=Goss-p1/>
Similarly all the polynomials of the form
<math display=block>\tau_p^n(x) = x^{p^n}</math>
are additive, where <math>n</math> is a non-negative integer.<ref name=Goss-p1/>
The definition makes sense even if <math>k</math> is a field of characteristic zero, but in this case the only additive polynomials are those of the form <math>ax</math> for some <math>a</math> in <math>k</math>.Template:Citation needed
The ring of additive polynomials
It is quite easy to prove that any linear combination of polynomials <math>\tau_p^n(x)</math> with coefficients in <math>k</math> is also an additive polynomial.<ref name=Goss-p1/> An interesting question is whether there are other additive polynomials except these linear combinations. The answer is that these are the only ones.<ref>Template:Harvnb</ref>
One can check that if <math>P(x)</math> and <math>M(x)</math> are additive polynomials, then so are <math>P(x)+M(x)</math> and <math>P(M(x))</math>. These imply that the additive polynomials form a ring under polynomial addition and composition. This ring is denoted<ref>Equivalently, Template:Harvnb defines <math>k\{ \tau_p\}</math> to be the ring generated by <math>\tau_p^n(x)</math> and then proves (p. 3) that it consists of all additive polynomials.</ref>
<math display=block>k\{ \tau_p\}.</math>
This ring is not commutative unless <math>k</math> is the field <math>\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}</math> (see modular arithmetic).<ref name=Goss-p1/> Indeed, consider the additive polynomials <math>ax</math> and <math>x^p</math> for a coefficient <math>a</math> in <math>k</math>. For them to commute under composition, we must have
<math display=block>(ax)^p = ax^p,\,</math>
and hence <math>a^p-a=0</math>. This is false for <math>a</math> not a root of this equation, that is, for <math>a</math> outside <math>\mathbb{F}_p.</math><ref name=Goss-p1/>
The fundamental theorem of additive polynomials
Let <math>P(x)</math> be a polynomial with coefficients in <math>k</math>, and <math>\{w_1,\dots,w_m\}\subset k</math> be the set of its roots. Assuming that the roots of <math>P(x)</math> are distinct (that is, <math>P(x)</math> is separable), then <math>P(x)</math> is additive if and only if the set <math>\{w_1,\dots,w_m\}</math> forms a group with the field addition.Template:Sfn
See also
References
External links
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