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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]
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 ...
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 ...
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 ...
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 ...
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.
Above, we said that a free Boolean algebra is a Boolean algebra with a set of generators that behave a certain way; alternatively, one might start with a set and ask which algebra it generates. Every set X generates a free Boolean algebra FX defined as the algebra such that for every algebra B and function f : X → B, there is a unique Boolean ...
If f is such a homomorphism, the scalar multiplication is (r, x) ↦ f(r)x (here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by r ↦ r ⋅ 1 A. (See also § From ring homomorphisms below). Every ring is an associative Z-algebra, where Z denotes the ring of the integers.
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