22:
92:
590:
1058:
734:
1269:
812:
649:
901:
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.
1353:
1137:
316:
1192:
997:
895:
51:
102:
481:
1397:
73:
160:
1439:
1013:
132:
117:
139:
665:
1444:
1216:
396:
they are not, and the weaker condition is the "wrong" as it does not behave well. For example, over a field of order
146:
34:
756:
613:
128:
213:
44:
38:
30:
1306:
190:
244:
55:
1086:
832:
is a field of characteristic zero, but in this case the only additive polynomials are those of the form
255:
1165:
962:
859:
1449:
1288:
953:
608:
1356:
206:
1155:
1061:
949:
853:
153:
1412:
1393:
342:
1003:
473:
1368:
1415:
1300:
1433:
1373:
393:
210:
231:
182:
109:
91:
1385:
220:
1420:
652:
392:
in the field. For infinite fields the conditions are equivalent, but for
822:
585:{\displaystyle (a+b)^{p}=\sum _{n=0}^{p}{p \choose n}a^{n}b^{p-n}.}
329:. It is equivalent to assume that this equality holds for all
85:
15:
448:
in the field, but will usually not be (absolutely) additive.
1053:{\displaystyle \mathbb {F} _{p}=\mathbf {Z} /p\mathbf {Z} }
113:
1080:. For them to commute under composition, we must have
948:)). These imply that the additive polynomials form a
617:
1309:
1219:
1168:
1089:
1016:
965:
862:
759:
668:
616:
484:
258:
1271:
be the set of its roots. Assuming that the roots of
1347:
1263:
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991:
889:
806:
728:
643:
584:
310:
729:{\displaystyle (a+b)^{p}\equiv a^{p}+b^{p}\mod p}
547:
534:
43:but its sources remain unclear because it lacks
1264:{\displaystyle \{w_{1},\dots ,w_{m}\}\subset k}
1197:The fundamental theorem of additive polynomials
1390:Basic Structures of Function Field Arithmetic
1064:). Indeed, consider the additive polynomials
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8:
1342:
1310:
1252:
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982:
969:
118:introducing citations to additional sources
807:{\displaystyle \tau _{p}^{n}(x)=x^{p^{n}}}
750:Similarly all the polynomials of the form
644:{\displaystyle \scriptstyle {p \choose n}}
1336:
1317:
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1218:
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920:) are additive polynomials, then so are
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74:Learn how and when to remove this message
108:Relevant discussion may be found on the
1209:) be a polynomial with coefficients in
1348:{\displaystyle \{w_{1},\dots ,w_{m}\}}
356:is used for the weaker condition that
352:is used for the condition above, and
189:are an important topic in classical
7:
852:It is quite easy to prove that any
828:The definition makes sense even if
1132:{\displaystyle (ax)^{p}=ax^{p},\,}
625:
538:
337:in some infinite field containing
311:{\displaystyle P(a+b)=P(a)+P(b)\,}
14:
1187:{\displaystyle \mathbb {F} _{p}.}
992:{\displaystyle k\{\tau _{p}\}.\,}
1046:
1033:
890:{\displaystyle \tau _{p}^{n}(x)}
848:The ring of additive polynomials
101:relies largely or entirely on a
90:
20:
1158:of this equation, that is, for
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952:under polynomial addition and
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1:
460:is additive. Indeed, for any
468:in the algebraic closure of
1466:
1392:, 1996, Springer, Berlin.
1359:with the field addition.
1279:) are distinct (that is,
1150:= 0. This is false for
956:. This ring is denoted
29:This article includes a
1440:Algebraic number theory
191:algebraic number theory
58:more precise citations.
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993:
904:One can check that if
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659:, which implies that
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312:
1416:"Additive Polynomial"
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897:with coefficients in
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129:"Additive polynomial"
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817:are additive, where
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666:
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609:binomial coefficient
482:
256:
187:additive polynomials
114:improve this article
877:
774:
599:is prime, for all
350:absolutely additive
240:additive polynomial
1445:Modular arithmetic
1413:Weisstein, Eric W.
1345:
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1072:for a coefficient
1062:modular arithmetic
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989:
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854:linear combination
821:is a non-negative
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739:as polynomials in
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321:as polynomials in
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31:list of references
1002:This ring is not
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343:algebraic closure
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214:characteristic
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112:. Please help
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39:external links
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451:
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420: +
419:
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412:will satisfy
411:
408: −
407:
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400:any multiple
399:
395:
394:finite fields
391:
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371:
367:
364: +
363:
359:
355:
351:
348:Occasionally
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238:is called an
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131: –
130:
126:
125:Find sources:
119:
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105:
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103:single source
99:This article
97:
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78:
75:
67:
57:
53:
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40:
36:
32:
27:
18:
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1374:Additive map
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1147:
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243:
239:
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232:coefficients
227:
223:
216:
202:
200:
186:
180:
167:
157:
150:
143:
136:
124:
100:
70:
61:
50:Please help
42:
1450:Polynomials
1004:commutative
954:composition
183:mathematics
56:introducing
1434:Categories
1386:David Goss
1380:References
1142:and hence
603:= 1, ...,
440:) for all
384:) for all
247:polynomial
221:polynomial
197:Definition
140:newspapers
1421:MathWorld
1327:…
1289:separable
1256:⊂
1237:…
974:τ
865:τ
836:for some
762:τ
692:≡
653:divisible
572:−
512:∑
245:Frobenius
110:talk page
64:June 2011
1363:See also
1355:forms a
1303:the set
1291:), then
1162:outside
452:Examples
354:additive
170:May 2024
1006:unless
823:integer
607:−1 the
242:, or a
230:) with
154:scholar
52:improve
1396:
1213:, and
1154:not a
936:) and
912:) and
595:Since
249:, if
185:, the
156:
149:
142:
135:
127:
1357:group
1287:) is
1060:(see
211:prime
207:field
205:be a
161:JSTOR
147:books
37:, or
1394:ISBN
1201:Let
1156:root
1068:and
950:ring
743:and
464:and
444:and
424:) =
388:and
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333:and
325:and
219:. A
201:Let
133:news
1076:in
840:in
719:mod
655:by
651:is
404:of
345:.
234:in
209:of
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181:In
116:by
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771:n
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670:(
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508:=
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492:+
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470:k
466:b
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302:b
299:(
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287:a
284:(
281:P
278:=
275:)
272:b
269:+
266:a
263:(
260:P
236:k
228:x
226:(
224:P
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203:k
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168:(
158:·
151:·
144:·
137:·
120:.
106:.
77:)
71:(
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62:(
48:.
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