Problem

Source: India Postals 2015 Set 3

Tags: roots of unity, algebra



Show that there are no positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ such that $$(1+a_1 \omega)(1+a_2 \omega)(1+a_3 \omega)(1+a_4 \omega)(1+a_5 \omega)(1+a_6 \omega)$$is an integer where $\omega$ is an imaginary $5$th root of unity.