<Question>
Let $z$ be a complex number such that $z^5 = 1$ and $z \neq 1.$  Compute
\[z + \frac{1}{z} + z^2 + \frac{1}{z^2}.\]
</Question>

<think>
<Plan_1>
Okay, so I need to compute the value of z + 1/z + z² + 1/z² where z is a complex number satisfying z⁵ = 1 and z ≠ 1. Hmm, let's start by understanding the problem. 

First, z is a fifth root of unity, but not equal to 1.
The equation z⁵ = 1 has five solutions, which are the fifth roots of unity. These can be expressed using Euler's formula as e^(2πik/5) for k = 0, 1, 2, 3, 4.

<prompt_1.1> Identify the fifth roots of unity and explain why z is one of them excluding z = 1. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Since z ≠ 1, z must be one of the other four roots: e^(2πi/5), e^(4πi/5), e^(6πi/5), or e^(8πi/5). 

Since all roots satisfy z⁵ = 1, we know that z⁵ - 1 = 0, which factors as (z - 1)(z⁴ + z³ + z² + z + 1) = 0. Therefore, for z ≠ 1, the minimal polynomial is z⁴ + z³ + z² + z + 1 = 0. That might be useful.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
The expression we need to compute is S = z + 1/z + z² + 1/z². Let me see if I can simplify this.

<prompt_2.1> Simplify the expression S = z + 1/z + z² + 1/z² using the fact that z⁵ = 1. </prompt_2.1>

Let me check if this is correct. Let's substitute z = e^(2πi/5).

<prompt_2.2> Verify the simplified expression S = 2cos(2π/5) + 2cos(4π/5) by substituting z = e^(2πi/5) and z = e^(4πi/5). </prompt_2.2>

Alternatively, we can note that since z⁴ + z³ + z² + z + 1 = 0, we can express higher powers of z in terms of lower powers. Let's see if that helps.

<prompt_2.3> Use the minimal polynomial z⁴ + z³ + z² + z + 1 = 0 to express higher powers of z in terms of lower powers. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Since z is a fifth root of unity, z⁵ = 1, so 1/z = z⁴, and 1/z² = z³. Let me verify that: if z⁵ = 1, then z⁻¹ = z⁵⁻¹ = z⁴, similarly z⁻² = z⁵⁻² = z³. So yes, 1/z = z⁴ and 1/z² = z³. Therefore, the expression S becomes z + z⁴ + z² + z³. 

Wait, but the original expression is z + 1/z + z² + 1/z², which substitutes to z + z⁴ + z² + z³. So S = z + z² + z³ + z⁴. But from the equation z⁴ + z³ + z² + z + 1 = 0, we know that z + z² + z³ + z⁴ = -1. Therefore, S = -1? That seems too straightforward.
</execution_2.1>

<execution_2.2>
Then 1/z = e^(-2πi/5) = e^(8πi/5), z² = e^(4πi/5), and 1/z² = e^(6πi/5). So S would be e^(2πi/5) + e^(8πi/5) + e^(4πi/5) + e^(6πi/5). 

But e^(8πi/5) is the same as e^(-2πi/5) because 8πi/5 - 2πi = 8πi/5 - 10πi/5 = -2πi/5. Similarly, e^(6πi/5) is e^(6πi/5 - 2πi) = e^(-4πi/5). So S is the sum of e^(2πi/5) + e^(-2πi/5) + e^(4πi/5) + e^(-4πi/5). 

Each pair like e^(2πik/5) + e^(-2πik/5) is equal to 2cos(2πk/5). Therefore, S = 2cos(2π/5) + 2cos(4π/5).
</execution_2.2>

<execution_2.3>
First, note that 1/z = z⁴ because z⁵ = 1, so z⁴ = 1/z. Similarly, 1/z² = z³. Therefore, the expression S = z + z⁴ + z² + z³. 

But from the minimal polynomial, z⁴ + z³ + z² + z + 1 = 0, so z⁴ + z³ + z² + z = -1. Therefore, S = (z + z⁴ + z² + z³) = -1. Wait, so does that mean the answer is -1?
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2:
But wait, from the equation z⁴ + z³ + z² + z + 1 = 0, we have z⁴ + z³ + z² + z = -1, so indeed S = -1. But when I substitute with the exponentials, I get 2cos(72°) + 2cos(144°).
Let me compute that numerically to check if it's equal to -1.

<prompt_3.1> Calculate the numerical value of 2cos(72°) + 2cos(144°) to verify if it equals -1. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
First, cos(72°) is approximately 0.3090, so 2cos(72°) ≈ 0.618. Then cos(144°) is cos(180° - 36°) = -cos(36°) ≈ -0.8090, so 2cos(144°) ≈ -1.618. Adding them together: 0.618 - 1.618 ≈ -1. So yes, that checks out numerically.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
But let me make sure I didn't skip any steps. The problem states z is a fifth root of unity not equal to 1, so all such z satisfy z⁴ + z³ + z² + z + 1 = 0.

<prompt_4.1> Confirm the solution by considering all fifth roots of unity and verifying the steps taken. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Therefore, z + z² + z³ + z⁴ = -1. Therefore, S = -1. 

But the problem is presented in a way that suggests it might require more steps, but perhaps this is sufficient. Let me check if there's another approach to confirm. 

Alternatively, consider that since z⁵ = 1, then z ≠ 0, so we can multiply the expression by z without issues. Let's see:

Compute S = z + 1/z + z² + 1/z². Multiply both sides by z:



Alternatively, if we didn't recall that trick, we could note that since z is a root of z⁴ + z³ + z² + z + 1 = 0, then z² + 1/z² can be expressed in terms of lower powers. Let's compute z² + 1/z².



Alternatively, using the identity for roots of unity. The sum over all roots of z^k for k from 0 to n-1 is 0. But here we are summing specific terms. 

Alternatively, note that the expression S = z + z² + z³ + z⁴. Let's denote this sum as S. Then, since z⁵ = 1, multiplying S by z gives zS = z² + z³ + z⁴ + z⁵. But z⁵ = 1, so zS = z² + z³ + z⁴ + 1. 

But from the equation z⁴ + z³ + z² + z + 1 = 0, we have z⁴ + z³ + z² + z = -1. Therefore, zS = (z² + z³ + z⁴) + z = (S - z) + z = S. Wait, that seems confusing. Let me write it again.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
We have S = z + z² + z³ + z⁴.

<prompt_5.1> Re-evaluate the expression S = z + z² + z³ + z⁴ using the minimal polynomial and verify the result. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
Multiply by z: zS = z² + z³ + z⁴ + z⁵. But z⁵ = 1, so zS = z² + z³ + z⁴ + 1. 

But from the equation z⁴ + z³ + z² + z + 1 = 0, we can write z⁴ + z³ + z² + 1 = -z. Therefore, zS = -z. 

Divide both sides by z (since z ≠ 0): S = -1. 

So that's another way to confirm it. Therefore, regardless of the approach, S = -1. 

Therefore, the answer should be \boxed{-1}.
</execution_5.1>
</Execution_5>

<Plan_6>
Based on execution_5.1:
**Final Answer**
\boxed{-1}
</Plan_6>

<Execution_6>
No prompt to execute.
</Execution_6>

<Final_answer>
Therefore, the final answer is $\boxed{-1}$.
</Final_answer>
</think>