Concepts inFast fourier transforms over poor fields
Fast Fourier transform
A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory; this article gives an overview of the available techniques and some of their general properties, while the specific algorithms are described in subsidiary articles linked below.
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Field (mathematics)
In abstract algebra, a field is a ring whose nonzero elements form a commutative group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth.
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Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of a linear operator.
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Finite field
In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, coding theory and Quantum error correction. The finite fields are classified by size; there is exactly one finite field up to isomorphism of size p for each prime p and positive integer k.
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Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients (or equivalently¿by clearing denominators¿with integer coefficients). Numbers such as ¿ that are not algebraic are said to be transcendental; almost all real and complex numbers are transcendental. (Here "almost all" has the sense "all but a countable set"; see Properties below.)
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Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function (which is often a function in the time domain). The DFT requires an input function that is discrete. Such inputs are often created by sampling a continuous function, such as a person's voice.
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Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence If G is an extension of Q by N, then G is a group, N is a normal subgroup of G and the quotient group G/N is isomorphic to group Q.
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