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Question

Let ω1 be a cube root of unity. Then the minimum of the set
{|a+bω+cω2|2:a,b,c distinct non-zero integers} equals

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Solution

|a+bω+cω2|2
=(a+bω+cω2)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(a+bω+cω2)
=(a+bω+cω2)(a+bω2+cω)
=(a2+abω2+acω+abω+b2ω3+ bcω2+acω2+bcω4+c2ω3)
=(a2+ω3(b2+c2)+ab(ω2+ω)+ac(ω2+ω)+bc(ω2+ω4))
=a2+b2+c2abacbc
=12[(ab)2+(bc)2+(ca)2]
Let a>b>c|ab|1,|bc|1,|ac|2
12[1+1+4]
So, minimum of set |a+bω+cω2|2=3

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