任給定一有限體上次數大於1的多項式M，此篇文章主要是估計:所有次數小於M，與M互質的多項式a，跟M做輾轉相除法所需的平均次數。

Let M be a monic polynomial over some finite fields. For polynomials a with deg a<deg M and (a,M)=1, we estimate the average value of the Euclidean Algorithm.

[1] H. Heilbronn,
On the average length of a class of finite continued fractions,
{\it Abhandlungen aus Zahlentheorie und Analysis,} VEB Deutsher Verlag, Berlin 1968.
[2] R. Lidl and H. Niederreiter,
{\it Finite Fields},
Cambridge University Press (1997).
[3] M. Rosen,
{\it Number Theory in Function Fields},
Springer-Verlag, GTM 210 (2002).
[4] Chih-Nung, Hsu,
A Polynomial Additive Divisor Problem.
[5] J. D. Dixon,
The number of steps in the Euclidean algorithm,
{\it J. Number Theory} 2 (1970), 414--422.
[6] T. Tonkov,
On the average length of a class of finite continued fractions,
{\it Acta Arith.} 26 (1974), 47--57.
[7] J. W. Porter,
On a theorem of Heilbronn,
{\it Mathematika.} 22 (1975), 20--28.
[8] H. Davenport,
{\it Multiplicative Number Theory},
Springer-Verlag, GTM 74 (1980).
[9] G. H. Norton,
On the asymptotic analysis of the Euclidean algorithm,
{\it J. Symbolic Computation} 10 (1990) 53--58.
[10] G. W. Effinger and D. R. Hayes,
{\it Additive Number Theory of Polynomials
Over a Finite Field},
Clarendon Press Oxford (1991).