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Difference between revisions of "Primitive polynomial"

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A [[Polynomial|polynomial]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p0745901.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p0745902.png" /> is a unique factorization domain, whose coefficients do not have common factors. Any polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p0745903.png" /> can be written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p0745904.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p0745905.png" /> a primitive polynomial and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p0745906.png" /> the [[Greatest common divisor|greatest common divisor]] of the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p0745907.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p0745908.png" />, defined up to multiplication by invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p0745909.png" />, is called the content of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p07459010.png" />. Gauss' lemma holds: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p07459011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074590/p07459012.png" />. In particular, a product of primitive polynomials is a primitive polynomial.
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A [[Polynomial|polynomial]] $f(X) \in R[X]$, where $R$ is a unique factorization domain, whose coefficients do not have common factors. Any polynomial $g(X) \in R[X]$ can be written in the form $g(X) = c(g) f(X)$ with $f(X)$ a primitive polynomial and $c(g)$ the [[greatest common divisor]] of the coefficients of $g(X)$. The element $c(G) \in R$, defined up to multiplication by invertible elements of $R$, is called the content of the polynomial $g(X)$. Gauss' lemma holds: If $g_1(X), g_2(X) \in R[X]$, then $c(g_1g_2) = c(g_1)c(g_2)$. In particular, a product of primitive polynomials is a primitive polynomial.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR>
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====References====
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1''' , Wiley  (1982)  pp. 165</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Birkhoff,  S. MacLane,  "A survey of modern algebra" , Macmillan  (1953)  pp. 79</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1''' , Wiley  (1982)  pp. 165</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Birkhoff,  S. MacLane,  "A survey of modern algebra" , Macmillan  (1953)  pp. 79</TD></TR>
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Revision as of 19:17, 2 November 2014

A polynomial $f(X) \in R[X]$, where $R$ is a unique factorization domain, whose coefficients do not have common factors. Any polynomial $g(X) \in R[X]$ can be written in the form $g(X) = c(g) f(X)$ with $f(X)$ a primitive polynomial and $c(g)$ the greatest common divisor of the coefficients of $g(X)$. The element $c(G) \in R$, defined up to multiplication by invertible elements of $R$, is called the content of the polynomial $g(X)$. Gauss' lemma holds: If $g_1(X), g_2(X) \in R[X]$, then $c(g_1g_2) = c(g_1)c(g_2)$. In particular, a product of primitive polynomials is a primitive polynomial.

References

[1] O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)


Comments

References

[a1] P.M. Cohn, "Algebra" , 1 , Wiley (1982) pp. 165
[a2] G. Birkhoff, S. MacLane, "A survey of modern algebra" , Macmillan (1953) pp. 79
How to Cite This Entry:
Primitive polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_polynomial&oldid=14653
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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