If ωis a non-real cube root of unity then (1+ω)(1+ω^2)(1+ω^4)(1+ω^8)(1+ω^10)(1+ω^32) is equal to

If ωis a non-real cube root of unity then...

Key Concepts

Roots of Unity

Roots of unity are the complex solutions of the equation z? = 1. In this context, the cube roots of unity include one real root (1) and two non-real roots. The non-real cube roots exhibit special symmetry properties, including complex conjugation and rotational symmetry in the complex plane, which are often leveraged to simplify products and sums involving them.

Modular Arithmetic in Exponents

Since ? is a cube root of unity, it satisfies the relation ?³ = 1. This property allows us to reduce any exponent of ? modulo 3, simplifying expressions with seemingly high exponents. By applying modular arithmetic to the exponents, one can transform complex products into much simpler forms that are easier to evaluate.

Cyclotomic Polynomials

Cyclotomic polynomials are the minimal polynomials for primitive roots of unity, and for cube roots of unity, the polynomial x² + x + 1 serves this purpose. This concept helps in understanding the algebraic relationships among the roots and can be used to further simplify expressions involving sums and products of roots of unity.

Transcript

00:01 So in the given question we are told that omega is the non -real cube root, non -real cube root of, right? it is the non -real cube root of 1.

00:18 And we are told to evaluate, we are told to evaluate the value of an expression that is given as 1 plus omega, 1 plus omega times.

00:35 1 plus omega square times 1 plus omega raised to 4 plus sorry times 1 plus omega raised to 8 times 1 plus omega raised to 10 times 1 plus omega raised to 32 so this is the expression that we need to find a value you 4 right and we are given that omega is the complex cube root of 1 and there is a relation for a property of omega which says 1 plus omega plus omega square is equal to 0 so we are going to use this property as well as another property that since omega is the cube root of 1 we can say that omega cube is equal to 1 right so so any power of omega that is a multiple of 3 can be written as written as the value is equal to 1.

01:46 Right omega raised to any power that is a multiple of 3 would give us the value as 1.

01:52 And now we can write terms like omega raised to 4 as omega cube times omega and since omega cube is equal to 1, omega raised to 4 can be written as omega.

02:08 Then terms like omega raised to 8 can be written as omega raised to 6 times omega.

02:15 6 is a multiple of 3.

02:17 So omega raised to 6 is equal to 1.

02:20 So what we have is omega squared instead of omega raised to 8.

02:25 And similarly for omega raised to 10, it is, we can write it as omega raised to 9 times omega, which is equal to omega and for omega raised to 32 it would be omega raised to 30 times omega square which is equal to omega square right so let's make these substitutions in the above expression so what we have would be one plus omega times one plus omega square times 1 plus omega times 1 plus omega square times 1 plus omega square times 1 plus omega square...

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