What important cultural and scientific advancements do we owe to the arabian empire

The question is incorrectly asked. there was never an "Arab" Empire. There were several Islamic Caliphates that were officially Arabic but heavily influenced by Persians and even administered and ruled majoritarily by non-Arab Muslims such as North-African Berbers or the Turks. This question is most likely referring to the Abassid Caliphate that was the Caliphate that was the most powerful during the Golden Age of Islam. Abassids considered knowledge as one of the most important values and founded the House of Wisdom in Bagdad where many Muslim and especially non-Muslim scholars were tasked with gathering all the knowledge of the world and translating it into Arabic. This, saved many works written in Ancient Greek and classical Latin that had been deemed unfit for the Catholic Church as "pagan" books. However, knowledge from the Chinese, Indian, North African, Egyptian, Persian and Byzantine civilizations was also compiled. Among the major contributions of the Golden Age of Islam are:

- Algebra (Persian scientist Muhammad ibn Mūsā al-Khwārizmī, known in Latin as Algorithmi). Hindu-Arabic numerals that we use today and Algebra as well as several mathematical theorems.

- Medicine (Avicenna, Persian Scientist). Advances in the knowledge of pulmonary circulation.

- Astronomy (Al-Battani, Arab Astronomer, known in Latin as Albategnius). Celestial globes were invented by them.

- Literature (The Book of One Thousand and One Nights, several stories compiled through the centuries by several authors and scholars). Also, Muslims learned to make paper from the Chinese Han and introduced that technology to the Mediterranean and to Europe.

- Philosophy (Especially the works of Avicenna and Averroes). They played a major work in saving and transmitting the knowledge of Aristotle to the Muslim and Christian world.

- Architecture (the horseshoe arch, the arabesque decorations, muqarnas)

- Glass and Chrystal of several types.

- Painting (only some harem wall paintings have survived in the city of Samarra).

- Pottery (Arabesque-decorated)

- Geometry (girih and geometrical stained glass).

- Scientific method: Ibn al-Haytham (Alhazen) and Avicenna.