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Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional vector space. For a more general coefficient ring, the same construction works if we take the free module on n generators. The construction of the free algebra on E is functorial in nature and satisfies an appropriate universal property.
An algebraic equation is an equation involving polynomials, for which algebraic expressions may be solutions. If you restrict your set of constants to be numbers, any algebraic expression can be called an arithmetic expression. However, algebraic expressions can be used on more abstract objects such as in Abstract algebra.
Representation of the expression (8 − 6) × (3 + 1) as a Lisp tree, from a 1985 Master's Thesis [44] Except for numbers and variables, every mathematical expression may be viewed as the symbol of an operator followed by a sequence of operands. In computer algebra software, the expressions are usually represented in this way.
Graph of a sextic function, with 6 real roots (crossings of the x axis) and 5 critical points. Depending on the number and vertical locations of minima and maxima, the sextic could have 6, 4, 2, or no real roots. The number of complex roots equals 6 minus the number of real roots. In algebra, a sextic (or hexic) polynomial is a polynomial of ...
This means that there is an injective set map i : S → F S that satisfies the following universal property: given any algebra A in V and any map k : S → A, there exists a unique V-homomorphism f : F S → A such that f ∘ i = k. This generalizes the notions of free group, free abelian group, free algebra, free module etc. It has the ...
2. In geometry and linear algebra, denotes the cross product. 3. In set theory and category theory, denotes the Cartesian product and the direct product. See also × in § Set theory. · 1. Denotes multiplication and is read as times; for example, 3 ⋅ 2. 2. In geometry and linear algebra, denotes the dot product. 3.
The algebraic equations are the basis of a number of areas of modern mathematics: Algebraic number theory is the study of (univariate) algebraic equations over the rationals (that is, with rational coefficients). Galois theory was introduced by Évariste Galois to specify criteria for deciding if an algebraic equation may be solved in terms of ...
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
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