Final answer:
The key achievement from the Golden Age of Islam in the category of intellectual achievements is the development of algebra as a distinct mathematical discipline by scholars like al-Khwarizmī.
Step-by-step explanation:
The key achievement from the Golden Age of Islam in the category of intellectual achievements is (b) The development of algebra as a distinct mathematical discipline by scholars like al-Khwarizmī. During this time, scholars like al-Khwarizmī made important original contributions to knowledge and culture. They not only preserved classical Greek philosophy and science but also developed new mathematical concepts like algebra, which revolutionized mathematics and paved the way for modern scientific advancements.