Linear irreducible representations of finite groups over number fields

In this brief note, we present a method to construct explicitly all irreducible representations of finite groups over a number field, up to equivalence. As a byproduct, we describe how to find the irreducible representations of the generalized quaternion group \(Q(2^{n})\), of order \(2^{n}\), over a field \(L\), where \(\mathbb{Q}\leq L\leq \mathbb{Q}(\xi_{2^{n-1}})\) and \(\xi_{2^{n-1}}\) a primitive \(2^{n-1}\)-root of unity.