This group gathers results about polynomials and about the elementary algebra taught before abstract structures, including the roots of polynomial equations, the factoring of expressions, identities that hold for all numbers, and rules for counting or locating roots. It covers statements about the complex roots of real polynomials, criteria for irreducibility, formulas relating roots to coefficients, and the expansion of powers of sums. A theorem belongs here when its cited article places it in algebra at the level of polynomials, equations and algebraic identities.
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Polynomials and Elementary Algebra
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Gauss's Lemma (Polynomials) (Wikipedia)
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In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a theorem about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic).
Candido's identity (Wikipedia)
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Candido's identity, named after the Italian mathematician Giacomo Candido, is an identity for real numbers.
Fundamental Theorem of Algebra (Wikipedia)
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theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex
Rational Root Theorem (Wikipedia)
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of the equation are also called roots or zeros of the polynomial on the left side. The theorem states that each
Eisenstein's criterion - Wikipedia
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Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over
Descartes' Rule of Signs (Wikipedia)
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René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients.
Vieta's Formulas (Wikipedia)
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mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named
Newton's identities, Wikipedia
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formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric
Sturm's theorem (Wikipedia)
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In mathematics, the Sturm sequence of a univariate polynomial p is a sequence of polynomials associated with
Binomial Theorem (Wikipedia)
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In elementary algebra, the binomial theorem (or binomial expansion)
De Moivre's formula (Wikipedia)
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These can be used to give explicit expressions for the nth roots of unity, that is, complex numbers z such that
Difference of two squares (Wikipedia)
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In elementary algebra, a difference of two squares is one squared number
Complex conjugate root theorem - Wikipedia
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the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a
Budan's theorem - Wikipedia
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is a theorem for bounding the number of real roots of a polynomial in an interval, and computing the parity of this
Binet-Cauchy identity, Wikipedia
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In algebra, the Binet-Cauchy identity, named after Jacques
Vincent's theorem - Wikipedia
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Vincent-is a theorem that isolates the real roots of polynomials with rational coefficients. Even though Vincent's
Completing the Square (Wikipedia)
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In elementary algebra, completing the square is a technique for converting
Elementary Symmetric Polynomial (Wikipedia)
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in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric
Ax-Grothendieck Theorem (Wikipedia)
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theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and
Alexander-Hirschowitz theorem (Wikipedia)
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will impose independent types of conditions on homogenous polynomials and the hypersurface of dimension d with many
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