Explanation:
The key exponent of the modern idea of nation-state was G.W.F Hegel. Let's understand why Hegel is considered the key exponent of the modern idea of nation-state.

Hegel's concept of the state:
Hegel's concept of the state was different from the traditional view of the state. He believed that the state is a necessary condition for the realization of human freedom. According to Hegel, the state is not just a mechanism for maintaining law and order, but it is an ethical organism that embodies the will of the people.

The concept of nation-state:
Hegel's idea of the state was the foundation for the concept of nation-state. The nation-state is a political entity where the state and the nation are congruent. In other words, the state represents the will of the nation, and the nation is the source of the state's legitimacy.

Hegel's influence on nationalism:
Hegel's idea of the state and the nation-state had a profound influence on the development of nationalism. Nationalism is the belief that the nation is the most important political entity, and it should be the basis for the organization of the state. Hegel's concept of the state as an ethical organism that embodies the will of the people provided the foundation for the development of nationalism.

Conclusion:
In conclusion, Hegel is considered the key exponent of the modern idea of nation-state because his concept of the state and the nation-state had a profound influence on the development of nationalism. Hegel's idea of the state as an ethical organism that embodies the will of the people provided the foundation for the development of nationalism.