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  2. Septic equation - Wikipedia

    en.wikipedia.org/wiki/Septic_equation

    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]

  3. Function (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Function_(mathematics)

    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 ...

  4. Algebraic structure - Wikipedia

    en.wikipedia.org/wiki/Algebraic_structure

    Addition and multiplication are prototypical examples of operations that combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example, a + (b + c) = (a + b) + c and a(bc) = (ab)c are associative laws, and a + b = b + a and ab = ba are commutative laws. Many systems studied ...

  5. Universal algebra - Wikipedia

    en.wikipedia.org/wiki/Universal_algebra

    A homomorphism between two algebras A and B is a function h : A → B from the set A to the set B such that, for every operation f A of A and corresponding f B of B (of arity, say, n), h(f A (x 1, ..., x n)) = f B (h(x 1), ..., h(x n)). (Sometimes the subscripts on f are taken off when it is clear from context which algebra the function is from ...

  6. Function composition - Wikipedia

    en.wikipedia.org/wiki/Function_composition

    During the mid-20th century, some mathematicians adopted postfix notation, writing xf for f(x) and (xf)g for g(f(x)). [18] This can be more natural than prefix notation in many cases, such as in linear algebra when x is a row vector and f and g denote matrices and the composition is by matrix multiplication. The order is important because ...

  7. Algebraically closed field - Wikipedia

    en.wikipedia.org/wiki/Algebraically_closed_field

    No finite field F is algebraically closed, because if a 1, a 2, ..., a n are the elements of F, then the polynomial (x − a 1)(x − a 2) ⋯ (x − a n) + 1 has no zero in F. However, the union of all finite fields of a fixed characteristic p ( p prime) is an algebraically closed field, which is, in fact, the algebraic closure of the field F ...

  8. F-algebra - Wikipedia

    en.wikipedia.org/wiki/F-algebra

    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.

  9. Endomorphism ring - Wikipedia

    en.wikipedia.org/wiki/Endomorphism_ring

    Explicitly, given two such homomorphisms f and g, the sum of f and g is the homomorphism f + g : x ↦ f(x) + g(x). Under this operation End(A) is an abelian group. With the additional operation of composition of homomorphisms, End(A) is a ring with multiplicative identity. This composition is explicitly fg : x ↦ f(g(x)).

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