The Rashtrakuta dynasty was a prominent dynasty that ruled large parts of the Deccan region in India from the 6th to the 10th century. They were known for their patronage of art and architecture, particularly the rock-cut temples at Ellora.

The rock-cut Shiva temple at Ellora is a magnificent example of the architectural brilliance of the Rashtrakuta dynasty. It was built during the reign of King Krishna I, making option D the correct answer.

Here is a detailed explanation of why Krishna I is the correct answer:

1. Historical Context:
The Rashtrakuta dynasty was established by Dantidurga in the mid-8th century. Over the years, the dynasty expanded its territory and became powerful rulers in the Deccan region. The dynasty witnessed several notable kings who made significant contributions to art and architecture.

2. Ellora Caves:
The Ellora Caves are a UNESCO World Heritage Site located near Aurangabad in Maharashtra, India. These caves represent a unique combination of Hindu, Buddhist, and Jain rock-cut temples and monasteries. The caves were built between the 6th and 10th centuries, during the reign of various dynasties, including the Rashtrakutas.

3. Rock-cut Shiva Temple:
The rock-cut Shiva temple, also known as the Kailasanatha temple, is one of the most remarkable structures at Ellora. It is a monolithic temple carved out of a single rock, depicting Mount Kailash, the abode of Lord Shiva. The temple is dedicated to Lord Shiva and is a significant pilgrimage site for Shaivites.

4. King Krishna I:
Krishna I was one of the most notable rulers of the Rashtrakuta dynasty. He ascended the throne in 756 CE and ruled until 774 CE. During his reign, the Rashtrakutas reached the peak of their power and expanded their empire. Krishna I was known for his military conquests and patronage of art and architecture.

5. Patronage of Architecture:
Krishna I was a great patron of architecture and encouraged the construction of several temples and structures. The rock-cut Shiva temple at Ellora is considered one of his most significant architectural achievements. The temple represents the grandeur and artistic excellence of the Rashtrakuta dynasty.

In conclusion, the rock-cut Shiva temple at Ellora was built during the reign of King Krishna I of the Rashtrakuta dynasty. His patronage of art and architecture, including this magnificent temple, showcases the dynasty's cultural and architectural brilliance.