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Number Theory Seminar

Seminar information archive ~05/19Next seminarFuture seminars 05/20~

Date, time & place Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.)
Organizer(s) Naoki Imai, Shane Kelly

2013/07/03

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Takehito Yoshiki (University of Tokyo)
A general formula for the discriminant of polynomials over ¥mathbb{F}_2 determining the parity of the number of prime factors
(JAPANESE)
[ Abstract ]
In order to find irreducible polynomials over \\mathbb{F}_2 efficiently, the method using Swan's theorem is known. Swan's theorem determines the parity of the numberof irreducible factors of a polynomial f over \\mathbb{F}_2 with no repeated root, by using the discriminant {\\rm D}(\\tilde{f})\\pmod 8, where \\tilde{f} is a monic polynomial over \\mathbb{Z}_2 such that \\tilde{f}=f\\pmod 2. In the lecture, we will give the formula for the discriminant {\\rm D}(\\tilde{f}) \\pmod 8 for a polynomial f over \\mathbb{F}_2 with no repeated root. By applying this formula to various types of polynomials, we shall get the parity of the number of irreducible factors of them.