Polynomial ring

If there is a commutative ring with one , then the polynomial ring is the set of all polynomials with coefficients from the ring and the variables along with the usual addition and multiplication of polynomials. In abstract algebra, this must be distinguished from the polynomial functions , not least because different polynomials can induce the same polynomial function. ${\displaystyle R}$ ${\displaystyle 1}$ ${\displaystyle R[X]}$ ${\displaystyle R}$ ${\displaystyle X}$

Definitions

The polynomial ring R [ X ]

${\displaystyle R[X]}$ is the crowd

${\displaystyle R^{(\mathbb {N} _{0})}:=\left\{(a_{i})_{i\in \mathbb {N} _{0}}\mid a_{i}\in R,a_{i}=0\ \mathrm {f{\ddot {u}}r} {\text{ fast alle }}i\right\}}$

of the sequences in , in which almost all , i.e. all but a finite number of sequence members are the same . ${\displaystyle R}$ ${\displaystyle 0}$

The addition is carried out component by component:

${\displaystyle (a_{i})_{i\in \mathbb {N} _{0}}+(b_{i})_{i\in \mathbb {N} _{0}}:=(a_{i}+b_{i})_{i\in \mathbb {N} _{0}}}$

and the convolution of the sequences defines the multiplication

${\displaystyle (a_{i})_{i\in \mathbb {N} _{0}}\cdot (b_{i})_{i\in \mathbb {N} _{0}}:=\left(\sum _{i=0}^{k}a_{i}b_{k-i}\right)_{k\in \mathbb {N} _{0}}=\left(\sum _{i+j=k}a_{i}b_{j}\right)_{k\in \mathbb {N} _{0}}}$.

Through these links, a ring structure is defined in the space of finite sequences, this ring is referred to as . ${\displaystyle R[X]}$

In this ring it is defined as ${\displaystyle X\in R^{(\mathbb {N} _{0})}}$

${\displaystyle X=X^{1}:=(0,1,0,0,\dotsc )}$

and that is ${\displaystyle 1\in R^{(\mathbb {N} _{0})}}$

${\displaystyle 1:=X^{0}=(1,0,0,0,\dotsc )}$ .

From the definition of multiplication by convolution it follows that

${\displaystyle X^{k}:=\underbrace {X\cdot X\dotsm X} _{k{\text{-mal das }}X}=(\underbrace {0,0,\dotsc ,0} _{k{\text{ Nullen}}},1,0,0,\dotsc )}$

and in the right bracket there is a one exactly at the -th place, otherwise the sequence consists exclusively of zeros. ${\displaystyle (k+1)}$

With the generator , each element can now be made clear in the common Polynomschreibweise ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle R^{(\mathbb {N} _{0})}}$

${\displaystyle f=a_{0}+a_{1}X+a_{2}X^{2}+\dotsb +a_{n}X^{n}=\sum _{i=0}^{n}a_{i}X^{i}}$

being represented. The individual elements of the sequence are called the coefficients of the polynomial. ${\displaystyle a_{i}}$

This gives the polynomial ring over in the indefinite ${\displaystyle R[X]}$ ${\displaystyle R}$ ${\displaystyle X}$ .

The polynomial ring in several variables

The polynomial ring in several variables is defined recursively by:

${\displaystyle R[X_{1},\dotsc ,X_{n}]:=R[X_{1},\dotsc ,X_{n-1}][X_{n}]}$

Here we consider polynomials in the variable with coefficients from the polynomial ring , which is again defined in the same way. This can be continued until one has arrived at the definition of the polynomial ring in a variable. In , each element can be clearly identified as a ${\displaystyle X_{n}}$ ${\displaystyle R[X_{1},\dotsc ,X_{n-1}]}$ ${\displaystyle R[X_{1},\dotsc ,X_{n}]}$

${\displaystyle \sum _{k=(k_{1},\dotsc ,k_{n})\in \mathbb {N} _{0}^{n}}{a_{k}\,X_{1}^{k_{1}}\dotsm X_{n}^{k_{n}}}}$

write.

The polynomial ring in any number of indeterminates (with an index set ) can either be defined as the monoid ring over the free commutative monoid over or as the colimes of the polynomial rings over finite subsets of . ${\displaystyle J}$ ${\displaystyle J}$ ${\displaystyle J}$

The quotient field

Is a body, the designation for the quotient field is from , the rational function field . Similarly, the quotient field of a polynomial ring over several indeterminates is denoted by. ${\displaystyle K}$ ${\displaystyle K(X)}$ ${\displaystyle K[X]}$ ${\displaystyle K[X_{1},\dotsc ,X_{n}]}$ ${\displaystyle K(X_{1},\dotsc ,X_{n})}$

properties

Degree

The function

${\displaystyle {\begin{array}{rccl}\deg \colon &R[X]&\to &\quad \mathbb {N} _{0}\cup \{-\infty \}\\&f&\mapsto &{\begin{cases}\max \left\{k\in \mathbb {N} _{0}\mid a_{k}\neq 0\right\},&{\text{wenn }}f\neq 0\\-\infty ,&{\text{wenn }}f=0\end{cases}}\end{array}}}$

defines the degree of the polynomial in the indefinite . The following applies to the usual stipulations for comparison and addition: and applies to all . ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle -\infty }$ ${\displaystyle k\in \mathbb {N} _{0}}$ ${\displaystyle -\infty <k}$ ${\displaystyle -\infty +k=-\infty }$

The coefficient is called the leading coefficient of . ${\displaystyle a_{\deg(f)}}$ ${\displaystyle f\neq 0}$

It applies to everyone ${\displaystyle f,g\in R[X]}$

• ${\displaystyle \deg(f\cdot g)\leq \deg(f)+\deg(g)}$

(Does not contain zero divisors - more precisely: if the leading coefficients are not zero divisors - equality applies.) ${\displaystyle R}$

• ${\displaystyle \deg(f+g)\leq \max\{\deg(f),\deg(g)\}}$.

From this theorem of degrees it follows in particular that if is a body, the units correspond exactly to the polynomials with degree zero, and these are the constants not equal to zero. ${\displaystyle R}$

When a body is by the membership function to a Euclidean ring : there is a division with remainder , in which the radical has a smaller degree than the divisor. ${\displaystyle R}$ ${\displaystyle R[X]}$

Examples

1. Be the ring of whole numbers. Then and both are of degree 1. The product has degree 2, as is also calculated from . ${\displaystyle R:=\mathbb {Z} }$ ${\displaystyle f:=1+2X\neq 0}$ ${\displaystyle g:=1+3X\neq 0}$ ${\displaystyle f\cdot g=1+5X+6X^{2}}$ ${\displaystyle \operatorname {deg} (f\cdot g)=\operatorname {deg} (f)+\operatorname {deg} (g)}$
2. Let the remainder class ring modulo 6 (a ring with the non-trivial zero divisors 2 and 3) and as above and . Both are and also here of grade 1. But has grade 1 and . ${\displaystyle R:=\mathbb {Z} /6\mathbb {Z} }$ ${\displaystyle f:=1+2X}$ ${\displaystyle g:=1+3X}$ ${\displaystyle \not \equiv 0\mod 6}$ ${\displaystyle f\cdot g=1+5X+6X^{2}\equiv 1+5X\mod 6}$ ${\displaystyle 1=\operatorname {deg} (f\cdot g)<\operatorname {deg} (f)+\operatorname {deg} (g)=2}$

Degree theorem for polynomials in several variables

With a monom

${\displaystyle a_{k_{1},\dotsc ,k_{n}}\,X_{1}^{k_{1}}\dotsm X_{n}^{k_{n}}}$

one defines the sum of the exponents

${\displaystyle k_{1}+\dotsb +k_{n}}$

as the total degree of monomial, if . The degree of the non-vanishing polynomial ${\displaystyle a_{k_{1},\dotsc ,k_{n}}\,\neq 0}$ ${\displaystyle d}$

${\displaystyle \sum _{k=(k_{1},\dotsc ,k_{n})\in \mathbb {N} _{0}^{n}}{a_{k}\,X_{1}^{k_{1}}\dotsm X_{n}^{k_{n}}}}$

in several variables is defined as the maximum total degree of the (non-vanishing) monomials. A sum of monomials of the same total degree is a homogeneous polynomial . The sum of all monomials of degree , i.e. i. the maximum homogeneous subpolynomial of maximum degree plays the role of the leading coefficient (related to all variables together). (The leading coefficient of a single indeterminate is a polynomial in the other indeterminate.) ${\displaystyle d}$

The law of degrees also applies to polynomials in several variables.

Elementary operations, polynomial algebra

In polynomial notation, addition and multiplication for elements and the polynomial ring look like this: ${\displaystyle \textstyle f=\sum _{i=0}^{m}f_{i}X^{i}}$ ${\displaystyle \textstyle g=\sum _{i=0}^{n}g_{i}X^{i}}$ ${\displaystyle R[X]}$

${\displaystyle f+g=\sum _{k=0}^{\max(m,n)}(f_{k}+g_{k})X^{k}}$,

${\displaystyle f\cdot g=\sum _{k=0}^{m+n}\left(\sum _{i+j=k}f_{i}\cdot g_{j}\right)X^{k}}$

The polynomial ring is not only a commutative ring, but also a module over , whereby the scalar multiplication is defined by term. With that even a commutative associative algebra is over . ${\displaystyle R[X]}$ ${\displaystyle R}$ ${\displaystyle R[X]}$ ${\displaystyle R}$

Homomorphisms

If and are commutative rings with and is a homomorphism , then is too ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle 1}$ ${\displaystyle \varphi \colon A\to B}$

${\displaystyle {\tilde {\varphi }}\colon A[X]\to B[X],\quad \sum _{i=1}^{n}{a_{i}X^{i}}\,\mapsto \,\sum _{i=1}^{n}\varphi (a_{i})X^{i}}$ a homomorphism.

If and are commutative rings with and is a homomorphism, then there is an unambiguous homomorphism for each , which is restricted to equal and for which applies, namely . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle 1}$ ${\displaystyle \varphi \colon A\to B}$ ${\displaystyle b\in B}$ ${\displaystyle \phi _{b}\colon A[X]\to B}$ ${\displaystyle A}$ ${\displaystyle \varphi }$ ${\displaystyle \phi _{b}(X)=b}$ ${\displaystyle \phi _{b}\left(\sum {a_{i}X^{i}}\right)=\sum {\varphi (a_{i})b^{i}}}$

Algebraic properties

If there is a commutative ring with , then: ${\displaystyle R}$ ${\displaystyle 1}$

• Is zero divisor free , so too . ${\displaystyle R}$ ${\displaystyle R[X]}$
• Is factorial , so also ( Lemma von Gauss ) ${\displaystyle R}$ ${\displaystyle R[X]}$
• If there is a body, then it is Euclidean and therefore a main ideal ring. ${\displaystyle R}$ ${\displaystyle R[X]}$
• If Noetherian , then the following applies to the dimension of the polynomial ring in a variable over : ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle \dim(R[X])=\dim(R)+1}$
• If noetherian , then the polynomial ring with coefficients is noetherian. ( Hilbert's basic set ) ${\displaystyle R}$ ${\displaystyle R[X_{1},\dotsc ,X_{n}]}$ ${\displaystyle R}$
• If an integrity ring is and , then has a maximum of zeros. This is generally wrong about non-integrity rings. ${\displaystyle R}$ ${\displaystyle 0\neq f\in R[X]}$ ${\displaystyle f}$ ${\displaystyle \deg(f)}$
• A polynomial is invertible in if and only if it is invertible and all other coefficients are nilpotent in . In particular, a polynomial over an integrity ring can be inverted if and only if it is a constant polynomial , one unit being in . ${\displaystyle f=a_{n}X^{n}+\dotsb +a_{0}\in R[X]}$ ${\displaystyle R[X]}$ ${\displaystyle a_{0}}$ ${\displaystyle R}$ ${\displaystyle f\in R[X]}$ ${\displaystyle R}$ ${\displaystyle a_{0}}$ ${\displaystyle a_{0}}$ ${\displaystyle R}$

Polynomial function and insertion homomorphism

→ Main article : Theorem about the establishment homomorphism

Is

${\displaystyle f=a_{0}+a_{1}X+\dotsb +a_{n}X^{n}}$

a polynomial is what is called ${\displaystyle R[X]}$

${\displaystyle f_{R}\colon R\to R,\quad x\mapsto f_{R}(x)=a_{0}+a_{1}x+\dotsb +a_{n}x^{n}}$

the associated polynomial function . More generally defines a polynomial function for each ring homomorphism (in a commutative ring with 1). The index is often omitted. ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle \phi \colon R\to S}$ ${\displaystyle S}$ ${\displaystyle f_{S}\colon S\to S,\ x\mapsto f_{S}(x).}$

Conversely, polynomial rings over a commutative ring with 1 have the following universal property : ${\displaystyle R[X]}$ ${\displaystyle R}$

Given a ring (commutative with 1), a ring homomorphism and a , there is exactly one homomorphism with , so that is a continuation of , so it holds. ${\displaystyle S}$ ${\displaystyle \phi \colon R\to S}$ ${\displaystyle s\in S}$ ${\displaystyle \Phi \colon R\left[X\right]\to S}$ ${\displaystyle \Phi (X)=s}$ ${\displaystyle \Phi }$ ${\displaystyle \phi }$ ${\displaystyle \Phi \mid _{R}=\phi }$

This property is called “universal” because it uniquely defines the polynomial ring except for isomorphism . ${\displaystyle R[X]}$

The homomorphism

${\displaystyle \Phi \colon a_{0}+a_{1}X+a_{2}X^{2}+\dotsb +a_{n}X^{n}\longmapsto a_{0}+a_{1}s+\dotsb +a_{n}s^{n},}$

is called the evaluation (homomorphism) for or establishment (homomorphism) of . ${\displaystyle s}$ ${\displaystyle s}$

Examples

• If we put and , the mapping is identical; . ${\displaystyle S=R[X]}$ ${\displaystyle s=X}$ ${\displaystyle \Phi _{X}\colon R[X]\to R[X],\ f\mapsto f_{R[X]}(X)=f}$ ${\displaystyle \Phi _{X}=\operatorname {Id} _{R[X]}}$
• If we consider a polynomial ring with additional indeterminates (see  polynomials with several variables ) as an extension of , we get, analogous to the construction from the previous example, the insertion homomorphism as a monomorphism of in , ${\displaystyle R[X,X_{1},X_{2},\dotsc ,X_{n}]}$ ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}}$ ${\displaystyle R[X]}$ ${\displaystyle \Phi _{X}\colon R[X]\to R[X,Y],\ f\mapsto f_{R[X,Y]}(X)=f}$ ${\displaystyle R[X]}$ ${\displaystyle R[X,X_{1},X_{2},\dotsc ,X_{n}].}$

Polynomial functions

If a ring is (commutative with 1), then the set of mappings of is also a ring in itself and according to the universal property there is a homomorphism ${\displaystyle R}$ ${\displaystyle Abb(R,R)}$ ${\displaystyle R}$

${\displaystyle \Phi \colon R\left[X\right]\to Abb(R,R)}$

with (the constant mapping ) for all and (the identity mapping ). ${\displaystyle \Phi (a)=c_{a}}$ ${\displaystyle a\in R}$ ${\displaystyle \Phi (X)=id_{R}}$

${\displaystyle {\overline {f}}:=\Phi (f)}$

is the polynomial function assigned to the polynomial. The homomorphism ${\displaystyle f}$

${\displaystyle f\to {\overline {f}}}$

is not necessarily injective, for example for and is the associated polynomial function . ${\displaystyle R=\mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle f=X^{2}+X\in R\left[X\right]}$ ${\displaystyle {\overline {f}}=0}$

Examples

A polynomial over a finite field

Since in the finite field , the unit group of cyclically with the order is valid for the equation . That is why the polynomial function is the polynomial ${\displaystyle \mathbb {F} _{q}}$ ${\displaystyle q-1}$ ${\displaystyle x\in \mathbb {F} _{q}}$ ${\displaystyle x^{q}=x}$ ${\displaystyle f_{\mathbb {F} _{q}}\colon \mathbb {F} _{q}\to \mathbb {F} _{q}}$

${\displaystyle f=X^{q}-X=\prod _{a\in \mathbb {F} _{q}}(X-a)\in \mathbb {F} _{q}[X]}$

the null function , although not the null polynomial. ${\displaystyle f}$

If a prime number, then this corresponds exactly to Fermat's small theorem . ${\displaystyle q}$

Polynomials with two variables

If or is a polynomial different from the zero polynomial, then the number of zeros is finite. In the case of polynomials with several indeterminates, the set of zeros can also be finite: ${\displaystyle f\in \mathbb {Z} [X]}$ ${\displaystyle f\in \mathbb {R} [X]}$ ${\displaystyle f}$

The polynomial has the zeros and in . ${\displaystyle f=((X-2)(X-3))^{2}+Y^{2}\in \mathbb {R} [X,Y]}$ ${\displaystyle (2,0)}$ ${\displaystyle (3,0)}$ ${\displaystyle \mathbb {R} ^{2}}$

But there can also be infinite sets of zeros:

The polynomial has the unit circle line as a set of zeros , which is a compact subset of . The polynomial also has an infinite set of zeros, namely the function graph of the normal parabola, which is not compact. ${\displaystyle f=X^{2}+Y^{2}-1\in \mathbb {R} [X,Y]}$ ${\displaystyle \{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle g=Y-X^{2}\in \mathbb {R} [X,Y]}$

The study of sets of roots of polynomial equations with several indeterminates led to the development of the mathematical branch of algebraic geometry .

Polynomials in the complex

Every complex polynomial of degree has exactly zeros if one counts every zero according to its multiplicity. In contrast , a zero is no longer called -fold if it is a divisor of . ${\displaystyle f\in \mathbb {C} [X]}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle z}$ ${\displaystyle k}$ ${\displaystyle (X-z)^{k}}$ ${\displaystyle f}$ ${\displaystyle (X-z)^{k+1}}$

In particular, this fundamental theorem of algebra also applies to real polynomials , if these are understood as polynomials in . For example, the polynomial has the zeros and , there and also , so holds . ${\displaystyle f\in \mathbb {R} [X]}$ ${\displaystyle \mathbb {C} [X]}$ ${\displaystyle X^{2}+1}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle -\mathrm {i} }$ ${\displaystyle \mathrm {i} ^{2}=-1}$ ${\displaystyle (-\mathrm {i} )^{2}=-1}$ ${\displaystyle X^{2}+1=(X+\mathrm {i} )(X-\mathrm {i} )}$

literature

• Siegfried Bosch : Algebra. 7th edition. Springer-Verlag, 2009, ISBN 3-540-40388-4 , doi: 10.1007 / 978-3-540-92812-6 .
• Serge Lang : Algebra. 3rd edition, Graduate Texts in Mathematics, Springer Verlag, 2005, ISBN 978-0387953854 .