Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree

Depends on

• The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution

Used by

• Certified complex-polynomial root isolation in exact algebraic arithmetic Algorithm
• Real bounded-polynomial QSP phase synthesis Algorithm
• Every finite subgroup of the unit group of an integral domain is cyclic Corollary
• The reduction of Φₙ is irreducible over F_q exactly when [q] generates (ℤ/n)^× Corollary
• Quasi-finite does not imply finite Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• Admissible polynomial pair for Hermitian-reflection QSP Definition
• An algebraically closed field: every nonconstant polynomial has a root in the field Definition
• Constant-coefficient linear recurrences, their starting index and their characteristic polynomial Definition
• Integral elements over a commutative ring and algebraic integers Definition
• Monomials, coefficients, degree in each variable and total degree in F[x₁,…,xₙ] Definition
• Multiplicity of a plane curve at a point Definition
• Plane projective curves and their components Definition
• Polynomials that split and splitting fields of a polynomial or a family of polynomials Definition
• Rational formal power series, proper presentations and reduced denominators Definition
• Resultant of two plane forms, viewed in one variable Definition
• Tangent cone and tangent lines at a point Definition
• The annihilator set Ann(T)={p∈ F[x]:p(T)=0}; once existence is proved, its unique monic generator μ_T is the minimal polynomial Definition
• The companion matrix of a monic polynomial Definition
• The cyclotomic polynomials Φₙ∈ℤ[t], defined by ∏_d∣ nΦ_d=tⁿ-1 Definition
• The discriminant of a monic polynomial as the coefficient expression of Δₙ² Definition
• The monic greatest common divisor of two polynomials over a field Definition
• The monic resultant Res(f,g) from the symmetric coefficient expression of ∏ᵢ g(xᵢ) Definition
• Normalization of a nodal affine plane curve Example
• Normalizing the cuspidal plane curve Example
• Normalizing the nodal plane curve Example
• The divisor-sum identity at q=2, n=3 finds exactly two monic irreducible cubics Example
• The four roots of t⁴+t+1 over F₂ are the Frobenius powers of any one of them Example
• The rational function field ℝ(t) ordered by the eventual sign is an ordered field, worked out Example
• Working the Hilbert basis construction on an ideal of ℤ[x] with non-monic stages Example
• Φ₁ through Φ₁₂ computed from the divisor recursion Example
• A nonnegative even polynomial has a bounded parity complement Lemma
• A polynomial vanishing at every tuple from an infinite subdomain is the zero polynomial Lemma
• A single cancellation step lowers the degree of a polynomial in an ideal once its leading coefficient lies in a realised stage Lemma
• Curves without a common component meet finitely often Lemma
• Finite-variable polynomial algebras over fields are Noetherian by finite generators Lemma
• For a finite group of ring automorphisms the orbit polynomial is monic over the invariant subring, so the ring is integral over its invariants Lemma
• If p is a prime not dividing n, a rational minimal polynomial of a primitive n-th root of unity also kills its p-th power Lemma
• Over a Noetherian ring, an ideal of R[x] is generated by finitely many polynomials realising generators of its stages up to the stabilisation degree Lemma
• Polynomial rings over normal domains are normal Lemma

…and 27 more results.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Thomas W. Judson, Abstract Algebra: Theory and Applications, Chapter 17.1 (standard reference, not scraped)
• Neil Donaldson, Math 120B Notes, Section 22 (standard reference, not scraped)