All branches of mathematics are of use today. Everything around you has a large dependency on Maths. Your computer is built on a whole range of mathematical principles in more ways than you can imagine. Algebra features amongst many aspects of mathematics, far too many to outline.
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It is al-Kwarizmi, he developed algebra as it is known today.
Islamic algebra is the same as regular algebra. just like Arabic numerals is the same as 1 2 3. why is algebra important today? there is no difference between these questions.
Digital logic IS hardware that implements Boolean algebra.
Algebra is used repeatedly in scientific fields such as astronomy, computer science, and so on. After all, algebra and science go hand in hand. If one was to research further into this, algebra is also used for basic math. For people who like to shop and find the best deals, algebra is used big time in shopping.
In the medical field algebra is used mostly for calculating dosages for medications and in giving the proper amount. Basic algebra is used to also calculate vitals.