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

• 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
• 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
• 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
• 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
• 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 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
• 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
• Reducing f modulo gᵢ(xᵢ)=∏_s∈ Sᵢ(xᵢ-s) lowers each deg_xᵢ below | Sᵢ|, preserves the values on the grid, and preserves any top-degree coefficient whose exponents stay below the grid sizes Lemma
• The leading coefficients of the degree-n elements of an ideal of R[x], together with 0, form an ideal of R, and these ideals ascend with n Lemma
• χ_A(x) is monic of degree n; for n≥1 its xⁿ⁻¹ coefficient is -tr(A) and its constant coefficient is (-1)ⁿ det(A), while χ_0×0=1 Lemma
• ∑_d∣ nd N_q(d)=qⁿ for the counts N_q(d) of monic irreducibles of degree d over F_q Proposition
• Degree inequalities for sums and products over a commutative ring Proposition
• Finitely supported coefficient sequences and trimmed finite coefficient lists define the same formal polynomials Proposition
• Φ_pʳ(t)=∑_k<pt^kpʳ⁻¹, and Φ_pʳ(t+1) is Eisenstein at p Proposition
• Φₙ is irreducible over K exactly when [K(ζₙ):K]=φ(n), exactly when the embedding into (ℤ/n)^× is onto Proposition
• A monic irreducible of degree d over F_q has the d distinct roots α,α^q,…,α^qᵈ⁻¹ Theorem
• C(x) is not a rational formal power series, so (Cₙ) satisfies no eventual constant-coefficient linear recurrence Theorem
• Division algorithm for polynomials over a field Theorem
• Division by a monic polynomial over a commutative ring Theorem
• Every odd-degree real polynomial has a real root Theorem
• For every n≥1 there are infinitely many primes p with p≡1 (mod n) Theorem
• For gcd(n,q)=1 the reduction of Φₙ in F_q[t] is a product of distinct monic irreducibles, each of degree the order of [q] modulo n Theorem
• Rational root theorem Theorem

…and 4 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)