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Algebra and Computation, Edition: version 25 Mar 1999 by Madhu Sudan

By Madhu Sudan

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We claim that f(x) is irreducible over F x]. This follows from the fact that if f(x) splits, then we can nd a smaller degree polynomial satisfying the above { and this contradicts the minimality of l. We have thus shown that for all 2 K there is a corresponding irreducible monic polynomial of degree d. But some of these polynomials may be the same { fortunately, we can account for all the repetitions by noting that each polynomial of degree d has at most d roots { hence a polynomial can repeat at most d times.

2 Multivariate Factoring We now present some thoughts on extending this to multivariate polynomials. 2 repeatedly starting from n variables down to n 1 and so on till we reach two variables when we can use the previous algorithm. D is a polynomial ring and hence is in nite and in particular the condition jRj > 4d2 will be satis ed. This procedure, however, is clearly not very e cient. An alternative procedure is sketched below. Let f 2 R x; y1; y2; : : :; yn ] be a polynomial that has to be factored.

This yields polynomials g~k and lk such that g(x; y) = g~k (x; y):lk (x; y) (mod y2k ), where g~k is monic in x, and satis es g~k = g0 (mod y). We claim g~k = gk . To do so we set ~hk (x; y) = lk (x; y)h(x; y) and then notice that f(x; y) = g(x; y)h(x; y) = g~k (x; y)lk (x; y)h(x; y) (mod y2k ) = g~k (x; y)h~ k (x; y) (mod y2k ): Additionally g~k is monic and equals g0 (mod y). Thus g~k and h~ k satisfy all the properties of the solution to the Hensel lifting procedure after k iterations. But the solution to the Hensel lifting is unique, implying gk = g~k .

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