Instruct: Given a question, retrieve relevant passages that answer the question.
Query: What is the origin of the word algebra?
The word algebra comes from the Arabic الجبر (al-jabr lit. "the reunion of broken parts") from the title of the book Ilm al-jabr wa'l-muḳābala by the Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the fifteenth century, from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century.[5]
In algebra, one often studies infinite algebras which are generated (under the finitary operations of the algebra) by finitely many elements. In this case, the elements of the algebra have a natural system of finite encoding as expressions in terms of the generators and operations. The word problem here is thus to determine, given two such expressions, whether they represent the same element of the algebra.
Lyndon words have an application to the description of free Lie algebras, in constructing a basis for the homogeneous part of a given degree; this was Lyndon's original motivation for introducing these words. Lyndon words may be understood as a special case of Hall sets.
If "A" and "B" are two unital algebras, then an algebra homomorphism formula_5 is said to be "unital" if it maps the unity of "A" to the unity of "B". Often the words "algebra homomorphism" are actually used in the meaning of "unital algebra homomorphism", so non-unital algebra homomorphisms are excluded.
Some areas of mathematics that fall under the classification abstract algebra have the word algebra in their name; linear algebra is one example. Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name.
The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product "X" ⧢ "Y" of two words "X", "Y": the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation.
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
The word problem on free Heyting algebras is difficult. 
The only known results are that the free Heyting algebra on one generator is infinite, and that the free complete Heyting algebra on one generator exists (and has one more element than the free Heyting algebra).
The word 'algebra' is used for various branches and structures of mathematics. For their overview, see Algebra.
The bare word "algebra" may refer to:
This isn't precisely what would correspond to the direct product of infinitely many copies of , one for each point in , because of the smoothness restriction. Instead, it can be thought of in terms of smooth map from to ; a smooth parametrized loop in , in other words. This is why it is called the loop algebra.
In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra.
The field "F" is algebraically closed if and only if it has no finite algebraic extension because if, within the previous proof, the word "algebraic" is replaced by the word "finite", then the proof is still valid.
The isomorphism of formula_91 and formula_136, "as associative algebras", leads to the concept of the algebra of symbols formula_170. This is the space of symmetric polynomials, endowed with a product, the formula_171, that places the algebraic structure of the Lie algebra onto what is otherwise a standard associative algebra. That is, what the PBW theorem obscures (the commutation relations) the algebra of symbols restores into the spotlight.
A theorem of Radford states that a shuffle algebra over a field of characteristic 0 can be viewed as a polynomial algebra over the Lyndon words. More precisely, let "A" be an alphabet, let "k" be a field of characteristic 0 (or, more general, a commutative ℚ-algebra), and let "R" be the free noncommutative "k"-algebra . The words over "A" can then be identified with the "noncommutative monomials" (i.e., products of the "x") in "R"; namely, we identify a word ("a","a"...,"a") with the monomial "x""x"..."x". Thus, the words over "A" form a "k"-vector space basis of "R". Then, a "shuffle product" is defined on "R"; this is a "k"-bilinear, associative and commutative product, which is denoted by ш, and which on the words can be recursively defined by
The "shuffle algebra" on the alphabet "A" is defined to be the additive group "R" endowed with ш as multiplication. Radford's theorem now states that the Lyndon words are algebraically independent elements of this shuffle algebra, and generate it; thus, the shuffle algebra is isomorphic to a polynomial ring over "k", with the indeterminates corresponding to the Lyndon words.
In algebra, a linear Lie algebra is a subalgebra formula_1 of the Lie algebra formula_2 consisting of endomorphisms of a vector space "V". In other words, a linear Lie algebra is the image of a Lie algebra representation.
The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. The word "FOIL" was originally intended solely as a mnemonic for high-school students learning algebra. The term appears in William Betz's 1929 text, "Algebra for Today", where he states:
... first terms, outer terms, inner terms, last terms. (The rule stated above may also be remembered by the word FOIL, suggested by the first letters of the words first, outer, inner, last.)
William Betz was active in the mathematics reform movement in the United States at that time, had written many texts on elementary mathematics topics and had "devoted his life to the improvement of mathematics education." 
As a single word without an article, "algebra" names a broad part of mathematics. As a single word with an article or in plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the author. Usually, the structure has an addition, multiplication, and a scalar multiplication (see Algebra over a field). When some authors use the term "algebra", they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations. With a qualifier, there is the same distinction: Without an article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves). With an article, it means an instance of some abstract structure, like a Lie algebra, an associative algebra, or a vertex operator algebra. Sometimes both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers.
The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.
The origins of algebraic geometry mostly lie in the study of polynomial equations over the real numbers. By the 19th century, it became clear (notably in the work of Jean-Victor Poncelet and Bernhard Riemann) that algebraic geometry was simplified by working over the field of complex numbers, which has the advantage of being algebraically closed. Two issues gradually drew attention in the early 20th century, motivated by problems in number theory: how can algebraic geometry be developed over any algebraically closed field, especially in positive characteristic? (The tools of topology and complex analysis used to study complex varieties do not seem to apply here.) And what about algebraic geometry over an arbitrary field?
In other words, a Banach *-algebra is a Banach algebra over formula_3 which is also a *-algebra.
For "R" a commutative ring, the free (associative, unital) algebra on "n" indeterminates {"X"...,"X"} is the free "R"-module with a basis consisting of all words over the alphabet {"X"...,"X"} (including the empty word, which is the unit of the free algebra). This "R"-module becomes an "R"-algebra by defining a multiplication as follows: the product of two basis elements is the concatenation of the corresponding words:
Nevertheless, the Hispano-Arabic hypothesis continues to have a presence in popular culture today.[85] It is the claim that algebraic x is the abbreviation of a supposed loanword from Arabic in Old Spanish. The theory originated in 1884 with the German orientalist Paul de Lagarde, shortly after he published his edition of a 1505 Spanish/Arabic bilingual glossary[86] in which Spanish cosa ("thing") was paired with its Arabic equivalent, شىء (shayʔ), transcribed as xei. (The "sh" sound in Old Spanish was routinely spelled x.) Evidently Lagarde was aware that Arab mathematicians, in the "rhetorical" stage of algebra's development, often used that word to represent the unknown quantity. He surmised that "nothing could be more natural" (Nichts war also natürlicher...) than for the initial of the Arabic word—romanized as the Old Spanish x—to be adopted for use in algebra.[87] A later reader reinterpreted Lagarde's conjecture as having "proven" the point.[88] Lagarde was unaware that early Spanish mathematicians used, not a transcription of the Arabic word, but rather its translation in their own language, "cosa".[89] There is no instance of xei or similar forms in several compiled historical vocabularies of Spanish.[90]
Monadic Boolean algebras form a variety. They are to monadic predicate logic what Boolean algebras are to propositional logic, and what polyadic algebras are to first-order logic. Paul Halmos discovered monadic Boolean algebras while working on polyadic algebras; Halmos (1962) reprints the relevant papers. Halmos and Givant (1998) includes an undergraduate treatment of monadic Boolean algebra.
A polynomial in variables vanishes at all points of a line passing through the origin if and only if it is homogeneous. In this case, one says that the polynomial "vanishes" at the corresponding point of P. This allows us to define a "projective algebraic set" in P as the set , where a finite set of homogeneous polynomials vanishes. Like for affine algebraic sets, there is a bijection between the projective algebraic sets and the reduced homogeneous ideals which define them. The "projective varieties" are the projective algebraic sets whose defining ideal is prime. In other words, a projective variety is a projective algebraic set, whose homogeneous coordinate ring is an integral domain, the "projective coordinates ring" being defined as the quotient of the graded ring or the polynomials in variables by the homogeneous (reduced) ideal defining the variety. Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties.