Step 1: Identify the major oil-producing regions of India.
The earliest and still one of the most significant oil-producing states in India is Assam. The first oil well of India was discovered in Digboi, Assam in 1889. Since then, several other oil fields have been developed in Assam.
Step 2: Major oil fields in Assam.
- Digboi, Naharkatiya, Moran, Rudrasagar, and Sibsagar are some of the most important oil fields.
- Assam accounts for a large share of India's crude oil production.
Step 3: Analyze the other options.
- (B) Rajasthan: Barmer oil fields are important but relatively new.
- (C) Bihar: No significant oil fields.
- (D) Tamil Nadu: Some offshore production (Cauvery basin) exists, but not as significant as Assam.
Step 4: Conclusion.
The state with the major oil fields historically and currently is Assam.
\[
\boxed{\text{Assam}}
\]