Leading coefficient

Leading coefficient
Tag	`0P05RJC5XA`
URI	`m2://0P05RJC5XA`
Kind	definition
Writing	W0 stub
MSC	13F20
Defines	Leading coefficient
Uses	Degree of a polynomial

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Definition

Let $R$ be a ring and let $f = a_0 + a_1 x + \dots + a_n x^n$ be a nonzero polynomial in $R[x]$ of degree $d$ . The leading coefficient of $f$ is $a_d$ . The leading coefficient of the zero polynomial is $0$ .

Remarks

The leading coefficient of $3x^4 - x + 2$ is $3$ . A polynomial with leading coefficient $1$ is monic. In an integral domain, the leading coefficient of $f g$ is the product of the leading coefficients of $f$ and $g$ .

Explanations

L1 human

The leading coefficient is the number in front of the highest power of $x$ . For large values of $x$ , the leading term $a_d x^d$ decides the size of a real polynomial.

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What this needs

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Group concept
Ring definition
Polynomial ring definition
Degree of a polynomial definition
Leading coefficient definition

What depends on this

39 nodes use this node, directly or through other nodes.

Uniqueness of finite fields of a given order 010PKCQPXP
Existence of splitting fields 01441KDGBY
Uniqueness of the algebraic closure 01VQ7C4FTB
Galois field 04SGPNT8R9
Uniqueness of splitting fields 069A3BC3EP
The minimal polynomial divides every polynomial vanishing at the element 06K7SARZKR
K[x]/(f) is a field iff f is irreducible 0AWMPX35CW
R[x] is an integral domain iff R is 0B5Z1EC66J
Order of the Galois group of a Galois extension equals its degree 0BDJBH8B4Z
Monic polynomial 0BDX4MBKT4
Finite subgroups of the multiplicative group of a field are cyclic 0CKGBEJX6B
Impossibility of doubling the cube 0CQ357M0QS
K[x] is a Euclidean domain 0DQG39K2NH
Galois extension 0DS2F7GK05
The multiplicative group of a finite field is cyclic 0EBZ3G455G
Impossibility of trisecting the angle 0EWXPZCSZ0
Minimal polynomial of an algebraic element 0GK1WB9R98
Factor theorem 0K3MWA7DMC
A polynomial over a field has at most as many roots as its degree 0KJVQV3E0F
Reduction modulo p test for irreducibility 0KWWEXW0RN
Adjoining a root of a polynomial 0M893X7YY7
Division algorithm for polynomials 0MES3KS0ZC
Separable extension 0N019F6RE9
Degree of a simple extension is the degree of the minimal polynomial 0N5QW9K7MH
The minimal polynomial is irreducible 0NFEG48RDR
Irreducibility of the cyclotomic polynomials over the rationals 0SK1VHZ5R2
The degree of a constructible number is a power of two 0SSSM1D475
Primitive element theorem 0SVX98HYV1
Fundamental theorem of Galois theory 0T1Q8D0FW8
Degree of a product 0T233AZ1GE
Existence of finite fields of every prime power order 0T8M3YC5RG
Remainder theorem 0TEDQH56RW
Every polynomial has a root in some extension field 0VG02M1FEZ
Polynomials of degree two or three are irreducible iff they have no roots 0VNE161QCK
Cyclotomic polynomial 0W0KVBM54S
Impossibility of squaring the circle 0X7CVDE3BG
Gauss–Wantzel theorem 0Y34JMJNE9
Primitive element of a finite field 0YAA9CJQXF
K(a) is isomorphic to K[x] modulo the minimal polynomial 0ZN39QDTP9