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  2. Irreducible polynomial - Wikipedia

    en.wikipedia.org/wiki/Irreducible_polynomial

    In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials.The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong.

  3. Irreducibility (mathematics) - Wikipedia

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

    In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial. In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module.

  4. Cyclotomic polynomial - Wikipedia

    en.wikipedia.org/wiki/Cyclotomic_polynomial

    In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any k < n. Its roots are all n th primitive roots of unity e 2 i π k n {\displaystyle e^{2i\pi {\frac {k}{n}}}} , where k runs over the positive integers less ...

  5. Eisenstein's criterion - Wikipedia

    en.wikipedia.org/wiki/Eisenstein's_criterion

    The fact that the polynomial after substitution is irreducible then allows concluding that the original polynomial is as well. This procedure is known as applying a shift . For example consider H = x 2 + x + 2 , in which the coefficient 1 of x is not divisible by any prime, Eisenstein's criterion does not apply to H .

  6. Minimal polynomial (field theory) - Wikipedia

    en.wikipedia.org/wiki/Minimal_polynomial_(field...

    The minimal polynomial f of α is irreducible, i.e. it cannot be factorized as f = gh for two polynomials g and h of strictly lower degree. To prove this, first observe that any factorization f = gh implies that either g ( α ) = 0 or h ( α ) = 0, because f ( α ) = 0 and F is a field (hence also an integral domain ).

  7. Algebraically closed field - Wikipedia

    en.wikipedia.org/wiki/Algebraically_closed_field

    The assertion "the polynomials of degree one are irreducible" is trivially true for any field. If F is algebraically closed and p(x) is an irreducible polynomial of F[x], then it has some root a and therefore p(x) is a multiple of x − a. Since p(x) is irreducible, this means that p(x) = k(x − a), for some k ∈ F \ {0} .

  8. Hilbert's irreducibility theorem - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_irreducibility...

    In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible.

  9. Algebraic variety - Wikipedia

    en.wikipedia.org/wiki/Algebraic_variety

    As g(x, y) is an absolutely irreducible polynomial, this is an algebraic variety. The set of its real points (that is the points for which x and y are real numbers), is known as the unit circle ; this name is also often given to the whole variety.