Why is there an "al" in algebra?

In Arabic, the definite article is always prefixed, never standing on its own as a word. Thus, the original Arabic word الجبر (transliterated al-jabr) became Latin algebra.

All three words have Arabic origins, but they entered English via other languages which had already imported the definite article with them.

The Arabic indefinite article is not recognized as such by the foreign listener. To a foreign speaker the 'al-' sounds like it is part of the original word, it is not obvious that it is an article, and so is not something that is translated. The entire sound is considered a new word.

That is the general rule for borrowing from another language.

In the particular instances you give, for 'al-' from Arabic, they are almost entirely borrowed from an intermediate language (Spanish, French or Latin) which already made the foreign (Arabic) article part of the word.

The name came from a book in arabic called "hisab aljaber wa almukabala" the word al-jaber means the balance. That book gave the idea of algebra. alchemy also came from arabic. algorithm was derived from the name of the person who wrote that book as well.