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## Sagot :

1.

**Rewrite the equation:**

[tex]\[ (1 + \omega^2)^4 - \omega = 0 \][/tex]

2.

**Introduce a new variable:**

Let's denote [tex]\( z = 1 + \omega^2 \)[/tex]. Then the equation becomes:

[tex]\[ z^4 - \omega = 0 \][/tex]

But we also know that [tex]\( z = 1 + \omega^2 \)[/tex], so we have a system of equations:

[tex]\[ z^4 = \omega \][/tex]

[tex]\[ z = 1 + \omega^2 \][/tex]

3.

**Combine the equations:**

Substitute [tex]\( \omega = z^4 \)[/tex] from the first equation into the second:

[tex]\[ z = 1 + (z^4)^2 \][/tex]

Simplifying, we get:

[tex]\[ z = 1 + z^8 \][/tex]

[tex]\[ z^8 + z - 1 = 0 \][/tex]

4.

**Solve for [tex]\( z \)[/tex]:**

Solve the polynomial [tex]\( z^8 + z - 1 = 0 \)[/tex].

5.

**Find [tex]\( \omega \)[/tex]:**

We need to revert back to [tex]\( \omega \)[/tex] using [tex]\( \omega = z^4 \)[/tex].

Solving this polynomial equation is quite complex, and we get a combination of real and complex roots. Let's list the roots for [tex]\( z \)[/tex]:

The roots of this equation would be complex numbers, and specifically, [tex]\(z\)[/tex] would be a complex number such that:

[tex]\[ z \approx -1/2 \pm \sqrt{3}i/2 \][/tex]

These solutions correspond to:

[tex]\[ z = -1/2 - \sqrt{3}i/2 \quad \text{and} \quad z = -1/2 + \sqrt{3}i/2 \][/tex]

Convert these complex roots back to [tex]\( \omega \)[/tex]:

[tex]\[ \omega = z^4 \][/tex]

So we get:

[tex]\[ \omega = \left(-\frac{1}{2} - \frac{\sqrt{3}}{2}i\right)^4 \quad \text{and} \quad \omega = \left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)^4 \][/tex]

6.

**Find the complex roots involving Root of Unity:**

Aside from these complex roots, we also notice that for higher-degree polynomials, SymPy provides roots in terms of `CRootOf`, showing the complexity and number of solutions.

To summarize, the solutions to the equation [tex]\((1 + \omega^2)^4 = \omega\)[/tex] are:

[tex]\[ \omega = -\frac{1}{2} - \frac{\sqrt{3}}{2}i, \quad -\frac{1}{2} + \frac{\sqrt{3}}{2}i, \quad \text{and complex roots represented as CRootOf} \][/tex]

Where the remaining solutions are:

[tex]\[ \omega = \text{CRootOf}(x^6 - x^5 + 4x^4 - 3x^3 + 5x^2 - 2x + 1, k) \][/tex]

for [tex]\(k = 0, 1, 2, 3, 4, 5\)[/tex].