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Parity is the property of an integer of whether it is even or odd; For more examples, see Category:Algebraic properties of elements. Of operations: associative property; commutative property of binary operations between real and complex numbers; distributive property; For more examples, see Category:Properties of binary operations.
An element x ∈ F is algebraic over E if it is a root of a polynomial with coefficients in E, that is, if it satisfies a polynomial equation. e n x n + e n−1 x n−1 + ⋯ + e 1 x + e 0 = 0, with e n, ..., e 0 in E, and e n ≠ 0. For example, the imaginary unit i in C is algebraic over R, and even over Q, since it satisfies the equation i 2 ...
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
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.
For example, in elementary arithmetic, one has (+) = + (). Therefore, one would say that multiplication distributes over addition . This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers , polynomials , matrices , rings , and fields .
The value y = a n x is an algebraic integer because it is a root of q(y) = a n − 1 n p(y /a n), where q(y) is a monic polynomial with integer coefficients. If x is an algebraic number then it can be written as the ratio of an algebraic integer to a non-zero algebraic integer. In fact, the denominator can always be chosen to be a positive integer.