Original in Hungarian; translated with Google translate; polished by myself.

Let $\varepsilon$ be a primitive seventh root of unity. Which integers are of the form $|\alpha|^2$, where $\alpha$ is an element of $\Bbb Q(\varepsilon)$, the seventh cyclotomic field?

(It is not clearly stated what $|\alpha|$ means; I think that we consider a fixed complex embedding of $\Bbb Q(\varepsilon)$ so that $|\alpha|$ means the complex absolute value of $\alpha$.)
To solve this problem, we need to determine which integers can be expressed as the squared absolute value of an element in the seventh cyclotomic field $\mathbb{Q}(\varepsilon)$, where $\varepsilon$ is a primitive seventh root of unity. 

1. **Primitive Root of Unity**:
   A primitive seventh root of unity $\varepsilon$ satisfies the equation:
   \[
   \varepsilon^7 = 1 \quad \text{and} \quad \varepsilon^k \neq 1 \quad \text{for} \quad 1 \leq k < 7.
   \]
   The minimal polynomial of $\varepsilon$ over $\mathbb{Q}$ is:
   \[
   \Phi_7(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1.
   \]

2. **Cyclotomic Field**:
   The field $\mathbb{Q}(\varepsilon)$ is the splitting field of $\Phi_7(x)$ and has degree 6 over $\mathbb{Q}$.

3. **Norm in Cyclotomic Field**:
   The norm of an element $\alpha \in \mathbb{Q}(\varepsilon)$, denoted $N_{\mathbb{Q}(\varepsilon)/\mathbb{Q}}(\alpha)$, is the product of all Galois conjugates of $\alpha$. For $\alpha = a_0 + a_1\varepsilon + a_2\varepsilon^2 + a_3\varepsilon^3 + a_4\varepsilon^4 + a_5\varepsilon^5$, the norm is:
   \[
   N_{\mathbb{Q}(\varepsilon)/\mathbb{Q}}(\alpha) = \prod_{\sigma \in \text{Gal}(\mathbb{Q}(\varepsilon)/\mathbb{Q})} \sigma(\alpha).
   \]

4. **Absolute Value**:
   The absolute value $|\alpha|$ in the complex embedding is the Euclidean norm:
   \[
   |\alpha| = \sqrt{\alpha \overline{\alpha}},
   \]
   where $\overline{\alpha}$ is the complex conjugate of $\alpha$.

5. **Norm and Absolute Value**:
   For $\alpha \in \mathbb{Q}(\varepsilon)$, the norm $N_{\mathbb{Q}(\varepsilon)/\mathbb{Q}}(\alpha)$ is a rational number, and it is related to the absolute value by:
   \[
   |\alpha|^2 = N_{\mathbb{Q}(\varepsilon)/\mathbb{Q}}(\alpha).
   \]

6. **Possible Norm Values**:
   Since $\alpha$ is an algebraic integer in $\mathbb{Q}(\varepsilon)$, its norm $N_{\mathbb{Q}(\varepsilon)/\mathbb{Q}}(\alpha)$ is an integer. Therefore, $|\alpha|^2$ must be a non-negative integer.

7. **Conclusion**:
   The integers that can be expressed as $|\alpha|^2$ for some $\alpha \in \mathbb{Q}(\varepsilon)$ are precisely the non-negative integers.

The final answer is $\boxed{\text{non-negative integers}}$