In a manner relating to or using algebra; by means of algebraic methods and calculations.
From algebraical plus the adverbial suffix '-ly,' which transforms adjectives into adverbs indicating manner or method. This form standardized in mathematical writing by the 18th century.
When mathematicians say they solved something 'algebraically,' they mean using symbols and equations rather than geometry or other methods—it's a shorthand for 'using abstract symbolic logic!'
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