Additive polynomial: Difference between revisions

Latest revision as of 00:56, 4 November 2025

Script error: No such module "Unsubst". In mathematics, the additive polynomials are an important topic in classical algebraic number theory.

Definition

Let $k$ be a field of prime characteristic $k$ . A polynomial $P (x)$ with coefficients in $k$ is called an additive polynomial, or a Frobenius polynomial, if

$P (a + b) = P (a) + P (b)$

as polynomials in $a$ and $b$ . It is equivalent to assume that this equality holds for all $a$ and $b$ in some infinite field containing $k$ , such as its algebraic closure.

Occasionally absolutely additive is used for the condition above, and additive is used for the weaker condition that $P (a + b) = P (a) + P (b)$ for all $a$ and $b$ in the field.^[1] For infinite fields the conditions are equivalent,Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters". 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 $q$ any multiple $P$ of $x^{q} - x$ will satisfy $P (a + b) = P (a) + P (b)$ for all $a$ and $b$ in the field, but will usually not be (absolutely) additive.

Examples

The polynomial $x^{p}$ is additive.^[1] Indeed, for any $a$ and $b$ in the algebraic closure of $k$ one has by the binomial theorem

$(a + b)^{p} = \sum_{n = 0}^{p} (\binom{p}{n}) a^{n} b^{p - n} .$

Since $p$ is prime, for all $n = 1, \dots, p - 1$ the binomial coefficient $(\binom{p}{n})$ is divisible by $p$ , which implies that

$(a + b)^{p} \equiv a^{p} + b^{p} mod p$

as polynomials in $a$ and $b$ .^[1]

Similarly all the polynomials of the form

$τ_{p}^{n} (x) = x^{p^{n}}$

are additive, where $n$ is a non-negative integer.^[1]

The definition makes sense even if $k$ is a field of characteristic zero, but in this case the only additive polynomials are those of the form $a x$ for some $a$ in $k$ .Script error: No such module "Unsubst".

The ring of additive polynomials

It is quite easy to prove that any linear combination of polynomials $τ_{p}^{n} (x)$ with coefficients in $k$ is also an additive polynomial.^[1] An interesting question is whether there are other additive polynomials except these linear combinations. The answer is that these are the only ones.^[2]

One can check that if $P (x)$ and $M (x)$ are additive polynomials, then so are $P (x) + M (x)$ and $P (M (x))$ . These imply that the additive polynomials form a ring under polynomial addition and composition. This ring is denoted^[3]

$k {τ_{p}} .$

This ring is not commutative unless $k$ is the field $𝔽_{p} = ℤ / p ℤ$ (see modular arithmetic).^[1] Indeed, consider the additive polynomials $a x$ and $x^{p}$ for a coefficient $a$ in $k$ . For them to commute under composition, we must have

$(a x)^{p} = a x^{p},$

and hence $a^{p} - a = 0$ . This is false for $a$ not a root of this equation, that is, for $a$ outside $𝔽_{p} .$ ^[1]

The fundamental theorem of additive polynomials

Let $P (x)$ be a polynomial with coefficients in $k$ , and ${w_{1}, \dots, w_{m}} \subset k$ be the set of its roots. Assuming that the roots of $P (x)$ are distinct (that is, $P (x)$ is separable), then $P (x)$ is additive if and only if the set ${w_{1}, \dots, w_{m}}$ forms a group with the field addition.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

References

↑ ^a ^b ^c ^d ^e ^f ^g Script error: No such module "citation/CS1".
↑ Script error: No such module "Footnotes".
↑ Equivalently, Script error: No such module "Footnotes". defines $k {τ_{p}}$ to be the ring generated by $τ_{p}^{n} (x)$ and then proves (p. 3) that it consists of all additive polynomials.

Script error: No such module "Check for unknown parameters".

External links

Script error: No such module "Template wrapper".

[Goss-p1-1] ↑ ^a ^b ^c ^d ^e ^f ^g Script error: No such module "citation/CS1".

[2] Script error: No such module "Footnotes".

[3] Equivalently, Script error: No such module "Footnotes". defines $k {τ_{p}}$ to be the ring generated by $τ_{p}^{n} (x)$ and then proves (p. 3) that it consists of all additive polynomials.

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-{{nofootnotes|date=June 2011}}
 {{one source |date=May 2024}}
 In [[mathematics]], the '''additive polynomials''' are an important topic in  classical [[algebraic number theory]].
 ==Definition==
-Let ''k'' be a [[field (mathematics)|field]] of [[prime number|prime]] [[characteristic (algebra)|characteristic]] ''p''. A [[polynomial]] ''P''(''x'') with [[coefficient]]s in ''k'' is called an '''additive polynomial''', or a '''[[Frobenius endomorphism|Frobenius]] polynomial''', if
+Let <math>k</math> be a [[field (mathematics)|field]] of [[prime number|prime]] [[characteristic (algebra)|characteristic]] <math>k</math>. A [[polynomial]] <math>P(x)</math> with [[coefficient]]s in <math>k</math> is called an '''additive polynomial''', or a '''[[Frobenius endomorphism|Frobenius]] polynomial''', if
-:<math>P(a+b)=P(a)+P(b)\,</math>
+<math display=block>P(a+b)=P(a)+P(b)</math>
-as polynomials in ''a'' and ''b''. It is equivalent to assume that this equality holds for all ''a'' and ''b'' in some infinite field containing ''k'', such as its [[algebraic closure]].
+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 ''P''(''a''&nbsp;+&nbsp;''b'') = ''P''(''a'')&nbsp;+&nbsp;''P''(''b'') for all ''a'' and ''b'' in the field. For infinite fields the conditions are equivalent, but for [[finite field]]s they are not, and the weaker condition is the "wrong" as it does not behave well. For example, over a field of order ''q'' any multiple ''P'' of ''x''<sup>''q''</sup>&nbsp;−&nbsp;''x'' will satisfy ''P''(''a''&nbsp;+&nbsp;''b'') = ''P''(''a'')&nbsp;+&nbsp;''P''(''b'') for all ''a'' and ''b'' in the field, but will usually not be (absolutely) additive.
+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>{{citation
+ | last = Goss | first = David | author-link = David Goss
+ | doi = 10.1007/978-3-642-61480-4
+ | isbn = 3-540-61087-1
+ | location = Berlin
+ | page = 1
+ | publisher = Springer
+ | title = Basic Structures of Function Field Arithmetic
+ | year = 1996}}</ref> For infinite fields the conditions are equivalent,{{sfn|Goss|1996|loc=Proposition 1.1.5|p=2}} but for [[finite field]]s 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 ''x''<sup>''p''</sup> is additive. Indeed, for any ''a'' and ''b'' in the algebraic closure of ''k'' one has by the [[binomial theorem]]
+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>(a+b)^p = \sum_{n=0}^p {p \choose n} a^n b^{p-n}.</math>
+<math display=block>(a+b)^p = \sum_{n=0}^p {p \choose n} a^n b^{p-n}.</math>
-Since ''p'' is prime,  for all ''n'' = 1, ..., ''p''−1 the [[binomial coefficient]] <math>\scriptstyle{p \choose n}</math> is [[divisible]] by ''p'', which implies that
+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>(a+b)^p \equiv a^p+b^p \mod p</math>
+<math display=block>(a+b)^p \equiv a^p+b^p \mod p</math>
-as polynomials in ''a'' and ''b''.
+as polynomials in <math>a</math> and <math>b</math>.<ref name=Goss-p1/>
 Similarly all the polynomials of the form
-:<math>\tau_p^n(x) = x^{p^n}</math>
+<math display=block>\tau_p^n(x) = x^{p^n}</math>
-are additive, where ''n'' is a non-negative [[integer]].
+are additive, where <math>n</math> is a non-negative [[integer]].<ref name=Goss-p1/>
-The definition makes sense even if ''k'' is a field of characteristic zero, but in this case the only additive polynomials are those of the form ''ax'' for some ''a'' in ''k''.{{citation needed|date=January 2016}}
+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>.{{citation needed|date=January 2016}}
 ==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 ''k'' is also an additive  polynomial. An interesting question is whether there are other additive  polynomials except these linear combinations. The answer is that these are the only ones.
+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>{{harvnb|Goss|1996|loc=Corollary 1.1.6|page=3}}</ref>
-One can check that if ''P''(''x'') and ''M''(''x'') are  additive polynomials, then so are ''P''(''x'')&nbsp;+&nbsp;''M''(''x'') and ''P''(''M''(''x'')). These imply that the  additive polynomials form a [[ring (mathematics)|ring]] under polynomial addition and [[function composition|composition]]. This ring is denoted
+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 (mathematics)|ring]] under polynomial addition and [[function composition|composition]]. This ring is denoted<ref>Equivalently, {{harvnb|Goss|1996|p=1}} 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>k\{ \tau_p\}.\,</math>
+<math display=block>k\{ \tau_p\}.</math>
-This ring is not [[commutative ring|commutative]] unless ''k'' is the field <math>\mathbb{F}_p = \mathbf{Z}/p\mathbf{Z}</math> (see [[modular arithmetic]]). Indeed, consider the additive polynomials ''ax'' and ''x''<sup>''p''</sup> for a coefficient ''a'' in ''k''. For them to commute under composition, we must have
+This ring is not [[commutative ring|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>(ax)^p = ax^p,\,</math>
+<math display=block>(ax)^p = ax^p,\,</math>
-and hence ''a''<sup>''p''</sup>&nbsp;−&nbsp;''a'' = 0.  This is false for ''a'' not a [[root of a function|root]] of this equation, that is, for ''a'' outside <math>\mathbb{F}_p.</math>
+and hence <math>a^p-a=0</math>.  This is false for <math>a</math> not a [[root of a function|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 ''P''(''x'') be a polynomial with coefficients in ''k'', and <math>\{w_1,\dots,w_m\}\subset k</math> be the set of its roots. Assuming that the roots of ''P''(''x'') are distinct (that is, ''P''(''x'') is [[separable polynomial|separable]]), then ''P''(''x'') is additive [[if and only if]] the set <math>\{w_1,\dots,w_m\}</math> forms a [[group (mathematics)|group]] with the field addition.
+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 polynomial|separable]]), then <math>P(x)</math> is additive [[if and only if]] the set <math>\{w_1,\dots,w_m\}</math> forms a [[group (mathematics)|group]] with the field addition.{{sfn|Goss|1996|loc=Theorem 1.2.1|p=4}}
 ==See also==
@@ Line 52: / Line 59: @@
 ==References==
-* [[David Goss]], ''Basic Structures of Function Field Arithmetic'', 1996, Springer, Berlin. {{ISBN|3-540-61087-1}}.
+{{reflist}}
 ==External links==

Additive polynomial: Difference between revisions

Latest revision as of 00:56, 4 November 2025

Contents

Definition

Examples

The ring of additive polynomials

The fundamental theorem of additive polynomials

See also

References

External links

Navigation menu

Additive polynomial: Difference between revisions

Latest revision as of 00:56, 4 November 2025

Definition

Examples

The ring of additive polynomials

The fundamental theorem of additive polynomials

See also

References

External links

Navigation menu

Search