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In mathematics, the idea of a free object is one of the basic concepts of abstract algebra.Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure.
Septic equations solvable by radicals have a Galois group which is either the cyclic group of order 7, or the dihedral group of order 14, or a metacyclic group of order 21 or 42. [1] The L(3, 2) Galois group (of order 168) is formed by the permutations of the 7 vertex labels which preserve the 7 "lines" in the Fano plane. [1]
Consider the endofunctor 1 + (−), i.e. F : Set → Set sending X to 1 + X, where 1 is a one-point set, a terminal object in the category. An algebra for this endofunctor is a set X (called the carrier of the algebra) together with a function f : (1 + X) → X. Defining such a function amounts to defining a point x ∈ X and a function X → X
For example, let f(x) = x 2 and g(x) = x + 1, then (()) = + and (()) = (+) agree just for = The function composition is associative in the sense that, if one of ( h ∘ g ) ∘ f {\displaystyle (h\circ g)\circ f} and h ∘ ( g ∘ f ) {\displaystyle h\circ (g\circ f)} is defined, then the other is also defined, and they are equal, that is, ( h ...
Going one step ahead of universal algebra, most algebraic structures are F-algebras. For example, abelian groups are F-algebras for the same functor F(G) = 1 + G + G×G as for groups, with an additional axiom for commutativity: m∘t = m, where t(x,y) = (y,x) is the transpose on GxG. Monoids are F-algebras of signature F(M) = 1 + M×M.
On the other hand, if F is not algebraically closed, then there is some non-constant polynomial p(x) in F[x] without roots in F. Let q(x) be some irreducible factor of p(x). Since p(x) has no roots in F, q(x) also has no roots in F. Therefore, q(x) has degree greater than one, since every first degree polynomial has one root in F.
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The letter may be followed by a subscript: a number (as in x 2), another variable (x i), a word or abbreviation of a word as a label (x total) or a mathematical expression (x 2i+1). Under the influence of computer science, some variable names in pure mathematics consist of several letters and digits.
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