Algebra (source code)
= Algebra
{wiki}
Algebra studies <equations>, <algebraic operations>, and the structures that make their rules work. Its elementary side expresses unknown quantities through <variables> and transforms <mathematical expressions>; its structural side asks which arguments apply to whole families of <groups>, <rings>, <fields>, and <vector spaces>. The useful step is identifying exactly which properties a calculation needs, so that it can be reused in a new setting.
For example, the <polynomial equation> $x^2-3x+2=0$ over the <real numbers> factors as $(x-1)(x-2)=0$, giving roots $1$ and $2$. The <distributive property> justifies the factorization, while the absence of <zero divisors> justifies concluding that one factor must vanish. Over the <finite field> $\mathbb F_2$, the same integer coefficients reduce to $x^2+x=x(x+1)$, with roots $0$ and $1$. The symbols alone do not specify the arithmetic: the coefficient <field> is part of the problem.
In a <ring> with <zero divisors>, a familiar step can fail. Modulo $8$, all four residue classes $1,3,5,7$ satisfy $x^2=1$. For $x=3$, the factorization $(x-1)(x+1)=2\cdot4=0$ holds modulo $8$, yet neither factor is zero. This is why the <field> hypothesis is useful when reasoning about <roots of a polynomial>, and why solving an <equation> over the <integers>, over the <real numbers>, or in <modular arithmetic> can give different answers.
The structure is also useful beyond scalar arithmetic. <Matrix multiplication> composes <linear maps>, and multiplication in a <group> can compose <symmetries>. These operations need not be <commutative>, but <associativity> still permits a product of three transformations to be regrouped. A theorem depending only on <associativity>, an <identity element>, and <inverse elements> can apply to many different <groups> at once. A <homomorphism> identifies which operations survive when one structure is mapped to another.
Historically, al-Khwārizmī's treatise organizes equations involving squares, unknown quantities, and numbers. Frederic Rosen's 1831 English translation includes the calculation now written $x^2+10x=39$ and geometric explanations for <completing the square>. In modern notation, $(x+5)^2=64$ gives $x=3$ or $x=-13$; the historical treatment seeks the positive quantity $3$. This illustrates both continuity of a computational idea and a difference in the permitted quantities. https://en.wikisource.org/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing/Compendium_on_calculating_by_completion_and_reduction[Al-Khwārizmī, Compendium on calculating by completion and reduction, Rosen's 1831 translation, pp. 5–11]{ref}
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