MathDB

8

Part of PEN Q Problems

Problems(1)

Q 8

Source:

5/25/2007
Show that a polynomial of odd degree 2m+12m+1 over Z\mathbb{Z}, f(x)=c2m+1x2m+1+⋯+c1x+c0,f(x)=c_{2m+1}x^{2m+1}+\cdots+c_{1}x+c_{0}, is irreducible if there exists a prime pp such that p∤c2m+1,p∣cm+1,cm+2,⋯ ,c2m,p2∣c0,c1,⋯ ,cm,  and  p3∤c0.p \not\vert c_{2m+1}, p \vert c_{m+1}, c_{m+2}, \cdots, c_{2m}, p^{2}\vert c_{0}, c_{1}, \cdots, c_{m}, \; \text{and}\; p^{3}\not\vert c_{0}.
algebrapolynomialanalytic geometrygraphing linesslopecombinatorial geometryPolynomials