Chapter 9: Polynomial Rings

9.1 Definitions and Basic Properties

Proposition 1: Let $R$ be an integral domain. Then

• $\mathrm{deg\;}p(x)q(x)=\mathrm{deg\;}p(x)+\mathrm{deg\;}q(x)$ if $p(x),q(x) \neq 0$.
• the units of $R[x]$ are just the units of $R$.
• $R[x]$ is an integral domain.

Proposition 2: Let $I$ be an ideal of the ring $R$ and let $(I)=I[x]$ denote the ideal of $R[x]$ generated by $I$ (polynomials with coefficients in $I$). Then $R[x]/(I)\cong(R/I)[x]$. In particular, if $I$ is a prime ideal of $R$ then $(I)$ is a prime ideal of $R[x]$.
Fact: If $I$ is as described above and $I$ is a maximal ideal of $R$, $(I,x)\subseteq R[x]$ is a maximal ideal too. $(I)$ is not necessarily maximal in $R[x]$.
Definition: The polynomial ring in variables $x_{1},\dots,x_{n}$ with coefficients in $R$ is denoted $R[x_{1},\dots,x_{n}]$ and defined inductively by $R[x_{1},\dots,x_{n}]=R[x_{1},\dots,x_{n-1}][x_{n}]$.
Fact: The degree of a multivariate monomial is the largest sum of the degrees of each variable taken individually. The degree of a multivariate polynomial is the maximum degree of its monomials. The subsum of a polynomial $f$ including only the monomial terms of degree $k$ is denoted the homogeneous component of $f$ of degree $k$. If $f$ has degree $d$, it is possible to express $f=\sum_{i=0}^{d}f_{i}$ for homogeneous components $f_{i}$ of degree $i$.

9.2 Polynomial Rings over Fields I

Theorem 3: Let $F$ be a field. The polynomial ring $F[x]$ is a Euclidean Domain. That is, letting $a(x),b(x)\in F[x]$ with $b(x) \neq 0$, then there exist unique $q(x),r(x)\in F[x]$ such that $a(x)=q(x)b(x)+r(x)$ with $r(x)=0$ or $\mathrm{deg\;}r(x)<\mathrm{deg\;}b(x)$. ($\mathrm{deg}$ is the norm).
Corollary 4: If $F$ is a field, then $F[x]$ is a PID and UFD (since it is already a Euclidean Domain).

9.3 Polynomial Rings that are Unique Factorization Domains

Proposition 5: (Gauss's Lemma) Let $R$ be a UFD with field of fractions $F$ and let $p(x)\in R[x]$. If $p(x)$ is reducible in $F[x]$ then $p(x)$ is reducible in $R[x]$. More precisely, if $p(x)=A(x)B(x)$ for some nonconstant polynomials $A(x),B(x)\in F[x]$, then $\exists r,s \in F$ with $r,s \neq 0$ such that $rA(x)=a(x)$ and $sB(x)=b(x)$, $a(x),b(x)\in R[x]$, and $p(x)=a(x)b(x)$ is a factorization in $R$.
Corollary 6: Let $R$ be a UFD, $F$ be its fields of fractions, and $p(x)\in R[x]$. Suppose the gcd of the coefficients of $p(x)$ is $1$. Then $p(x)$ is irreducible in $R[x]$ $\iff$ it is irreducible in $F[x]$. In particular, if $p(x)$ is monic and irreducible in $R[x]$ then $p(x)$ is irreducible in $F[x]$. An important implication of this is that a monic polynomial in $\mathbb{Z}[x]$ is irreducible $\iff$ it is irreducible over $\mathbb{Q}[x]$. (Can also be applied to polynomials in $\mathbb{Q}[x]$ with only integer coefficients).
Theorem 7: $R$ is a UFD $\iff$ $R[x]$ is a UFD.
Corollary 8: If $R$ is a UFD, then a multivariate polynomial ring with coefficients in $R$ is also a UFD.

9.4 Irreducibility Criteria

Proposition 9: Let $F$ be a field and let $p(x)\in F[x]$. Then $p(x)$ has a factor of degree $1$ $\iff$ $\exists\alpha \in F$ with $p(\alpha)=0$.
Proposition 10: A polynomial with $\mathrm{deg}=2$ or $3$ over a field $F$ is reducible $\iff$ it has a root in $F$.
Proposition 11: (Rational Root Theorem) Let $p(x)=a_{n}x_{n}+\dots+a_{0}\in \mathbb{Z}[x]$. If $\frac{r}{s}\in \mathbb{Q}$ in the lowest terms and $p\left( \frac{r}{s} \right) = 0$, then $r$ divides the constant term and $s$ divides the leading coefficient of $p(x)$: $r \mid a_{0}$ and $s\mid a_{n}$. In particular, if $p(x)$ is monic with integer coefficients and $p(d) \neq 0,\;\forall d\in \mathbb{Z}$ such that $d \mid a_{0}$, then $p(x)$ has no roots in $\mathbb{Q}$. More precisely, if $p(x)\in \mathbb{Z}[x]$ and is monic, all rational roots must be integers.
Proposition 12: Let $I$ be a proper ideal in the integral domain $R$ and let $p(x)$ be a nonconstant monic polynomial in $R[x]$. If the image of $p(x)$ in $(R/I)[x]$ cannot be factored in $(R/I)[x]$ into two polynomials of smaller degree, then $p(x)$ is irreducible in $R[x]$.
Proposition 13: (Eisenstein's Criterion) Let $P$ be a prime ideal of the integral domain $R$ and let $f(x)=x^{n}+a_{n-1}x^{n-1}+\dots+a_{0}\in R[x]$ ($n\geq1$). Suppose $a_{n-1},\dots,a_{0}\in P$ and $a_{0}\not\in P^{2}$. Then $f(x)$ is irreducible in $R[x]$.
Corollary 14: (Eisenstein's Criterion for $\mathbb{Z}[x]$) Let prime $p \in \mathbb{Z}$ and $f(x)=x^{n}+a_{n-1}x^{n-1}+\dots+a_{0}\in \mathbb{Z}[x]$, $n\geq1$. Suppose $p \mid a_{i},\ \forall i\in \{ 0,\dots,n-1 \}$ but $p^{2} \nmid a_{0}$. Then $f(x)$ is irreducible in both $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$.

9.5 Polynomial Rings over Fields II

Let $F$ be a field.
Proposition 15: The maximal ideals in $F[x]$ are the ideals $(f(x))$ generated by irreducible polynomials $f(x)$. In particular, $F[x]/(f(x))$ is a field $\iff$ $f(x)$ is irreducible.
Proposition 16: Let $g(x)$ be a nonconstant monic element of $F[x]$ and let $g(x)=f_{1}(x)^{n_{1}}\dots f_{k}(x)^{n_{k}}$ be its irreducible factorization. Then we have the following ring isomorphism: $F[x]/(g(x))\cong F[x]/(f_{1}(x)^{n_{1}})\times\dots \times F[x]/(f_{k}(x)^{n_{k}})$.
Proposition 17: If the polynomial $f(x)$ has roots $\alpha_{1},\dots,\alpha_{k}\in F$ (not necessarily distinct), then $(x-\alpha_{1})\dots(x-\alpha_{k}) \mid f(x)$. In particular, a univariate polynomial of degree $n$ over $F$ has at most $n$ roots in $F$, even with multiplicity.
Proposition 18: A finite subgroup of the multiplicative group of a field is cyclic. In particular, if $F$ is a finite field, then the multiplicative group $F^{\times}$ of nonzero elements of $F$ is a cyclic group.
Corollary 19: Let $p$ be a prime. The multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{\times}$ of nonzero residue classes mod $p$ is cyclic. (Since $(\mathbb{Z}/p\mathbb{Z})^{\times}$ is a finite field).
Corollary 20: Let $n\geq 2$ be an integer with factorization $n=p_{1}^{\alpha_{1}}..p_{r}^{\alpha_{r}}$ in $\mathbb{Z}$, where $p_{1},\dots p_{r}$ are distinct primes. Then, the following isomorphisms of multiplicative groups exist:

• $(\mathbb{Z}/n\mathbb{Z})^{\times }\cong(Z/p_{1}^{\alpha_{1}}\mathbb{Z})^{\times}\times\dots \times(\mathbb{Z}/p_{r}^{\alpha_{r}}\mathbb{Z})^{\times}$.
• $(\mathbb{Z}/2^{\alpha}\mathbb{Z})^{\times}\cong Z_{2}\times Z_{2^{\alpha-2}},\ \forall\alpha\geq2$.
• $(\mathbb{Z}/p^{\alpha}\mathbb{Z})^{\times}\cong Z_{p^{\alpha-1}(p-1)}$, for all odd primes $p$.

9.6 Polynomials in Several Variables over a Field and Gröbner Bases

Definition: A commutative ring $R$ with $1$ is called Noetherian if every ideal of $R$ is finitely generated.
Theorem 21: (Hilbert's Basis Theorem) If $R$ is a Noetherian ring then so is the polynomial ring $R[x]$.
Corollary 22: Every ideal in the polynomial ring $F[x_{1},\dots,x_{n}]$ with coefficients from a field $F$ is finitely generated.
Definition: A monomial ordering is a well ordering "$\geq$" on the set of monomials that satisfies $mm_{1}\geq mm_{2}$ whenever $m_{1}\geq m_{2}$ for monomials $m,m_{1},m_{2}$. Equivalently, a monomial ordering may be specified by defining a well ordering on the $n$-tuples $\alpha=(a_{1},\dots,a_{n})\in \mathbb{Z}^{n}$ of multidegrees of monomials $Ax_{1}^{a_{1}}\dots x_{n}^{a_{n}}$ that satisfies $\alpha+\gamma\geq\beta+\gamma$ if $\alpha\geq\beta$.
Definition: Fix a monomial ordering on the polynomial ring $F[x_{1},\dots,x_{n}]$.

• The leading term of a nonzero polynomial $f\in F[x_{1},\dots ,x_{n}]$, denoted $LT(f)$, is the monomial term of maximal order in $f$ and the leading term of $f=0$ is $0$. Define the multidegree of $f$, denoted $\partial(f)$, to be the multidegree of the leading term of $f$.
• If $I$ is an ideal in $F[x_{1},\dots,x_{n}]$, the ideal of leading terms, denoted $LT(I)$, is the ideal generated by the leading terms of all the elements in the ideal, i.e., $LT(I)=(LT(f)\mid f\in I)$.

Fact: $LT(fg)=LT(f)+LT(G)$. The ideal $LT(I)$ is also generated by monomials; such ideals are denoted as monomial ideals. A polynomial is contained in a monomial ideal $\iff$ each of its monomial terms is a multiple of one of the ideal's generators.
Definition: A Gröbner basis for an ideal $I$ in the polynomial ring $F[x_{1},\dots,x_{n}]$ is a finite set of generators $\{ g_{1},\dots,g_{m} \}$ for $I$ whose leading terms generate the ideal of all leading terms in $I$, i.e. $I=(g_{1},\dots,g_{m})$ and $LT(I)=(LT(g_{1}),\dots,LT(g_{m}))$.
Fact: Every element in $I$ is a linear combination of the generators that is not necessarily unique.
General Polynomial Division: Fix a monomial ordering on $F[x_{1},\dots,x_{n}]$ and suppose $g_{1},\dots,g_{m}$ is a set of nonzero polynomials in $F[x_{1},\dots,x_{n}]$. If $f\in F[x_{1},\dots,x_{n}]$, start with a set of quotients $q_{1},\dots,q_{m}$ and a remainder $r$ all equal to $0$. Then, recursively test if the leading term of $f$ is divisible by the any of the leading terms of the divisors $g_{1},\dots,g_{m}$ in that order.

• If $LT(g_{i})\mid LT(f)$ such that $LT(f)=a_{i}LT(g_{i})$, add $a_{i}$ to quotient $q_{i}$ and repeat the algorithm on $f-a_{i}g_{i}$.
• If the leading term of the dividend $f$ is not divisible by any of the leading terms $LT(g_{1}),\dots,LT(g_{m})$, add the leading term of $f$ to the remainder $r$ and repeat the algorithm on $f-LT(f)$.

Theorem 23: Fix a monomial ordering on $R=F[x_{1},\dots,x_{n}]$ and suppose $\{ g_{1},\dots,g_{m} \}$ is a Gröbner basis for the nonzero ideal $I$ in $R$. Then

• Every polynomial $f\in R$ can be written uniquely in the form $f_{I}+r$ where $f_{I}\in I$ and no nonzero monomial term of the remainder $r$ is divisible by any of the leading terms $LT(g_{1}),\dots,LT(g_{m})$.
• Both $f_{1}$ and $r$ can be computed by general polynomial division by $g_{1},\dots,g_{m}$ and are independent of the order in which these polynomials are used in the division.
• The remainder $r$ provides a unique representative for the coset of $f$ in $F[x_{1},\dots,x_{n}]/I$. In particular, $f\in I\iff r=0$.

Proposition 24: Fix a monomial ordering on $R=F[x_{1},\dots,x_{n}]$ and let $I$ be a nonzero ideal in $R$.

• If $g_{1},\dots,g_{m}\in I$ such that $LT(I)=(LT(g_{1}),\dots,LT(g_{m}))$.
• There exists a Gröbner basis for $I$.

Fact: Let $M$ be the monic LCM of $LT(f_{1})$ and $LT(f_{2})$. Define $S(f_{1},f_{2})=\frac{M}{LT(f_{1})}f_{1}-\frac{M}{LT(f_{2})}f_{2}$. Then $S(f_{1},f_{2})$ is a polynomial where the leading terms of $f_{1}$ and $f_{2}$ have been canceled.
Lemma 25: Suppose $f_{1},\dots,f_{m}\in F[x_{1},\dots,x_{n}]$ are polynomials with the same multidegree $\alpha$ and that the linear combination $h=a_{1}f_{1}+\dots+a_{m}f_{m}$ with constants $a_{i}\in F$ has a multidegree strictly smaller than $\alpha$. Then $h=\sum_{i=2}^{m}b_{i}S(f_{i-1},f_{i})$ for some constants $b_{i}\in F$.
Fact: For a fixed monomial ordering on $R=F[x_{1},\dots,x_{n}]$ and ordered set of polynomials $G=\{ g_{1},\dots, g_{m} \}$ in $R$, write $f\equiv r\; (\text{mod} \; G )$ if $r$ is the remainder of general polynomial division of $f\in R$ by $g_{1},\dots,g_{m}$ in that order.
Proposition 26: (Buchberger's Criterion) Let $R=F[x_{1},\dots,x_{n}]$ and fix a monomial ordering on $R$. If $I=\{ g_{1},\dots,g_{m} \}$ is a nonzero ideal in $R$, then $G=\{ g_{1},\dots,g_{m} \}$ is a Gröbner basis for $I$ $\iff S(g_{i},g_{j})\equiv 0\; (\text{mod} \; G ),\ 1\leq i<j\leq m$.
Buchberger's Algorithm: Let ideal $I=(g_{1},\dots,g_{m})$ and $G=\{ g_{1},\dots,g_{m} \}$. Let $S(g_{i},g_{j})$ have $r \neq 0$ for some ordered pair $(i,j)$. (Otherwise, if $S(g_{i},g_{j})=0,\ \forall (i,j)$, then $G$ is a Gröbner basis). Modify $G$ to be $G'=G\cup \{ r \}$. Iterate until done. Note that this will always terminate after a finite number of sets since there exist finitely many ordered pairs $(i,j)$.
Fact: If $G=\{ g_{1},\dots,g_{m} \}$ is a Gröbner basis for the ideal $I$ and $LT(g_{i}) \mid LT(g_{j})$ for some $j \neq i$, then $G-\{ g_{j} \}$ is still a valid Gröbner basis. Therefore, we may assume that the $LT(g_{i}),\ \forall i$ is monic. Then, a Gröbner basis $G=\{ g_{1},\dots,g_{m} \}$ for $I$ is called minimal if $LT(g_{i})$ is monic $\forall i$ and $LT(g_{i}) \nmid LT(g_{j}),\ \forall(i,j)$. (This is analogous to a linear independent basis in vector spaces). Note also that while a minimal Gröbner basis is not necessarily unique, the number of elements and their leading terms are unique.
Definition: Fix a monomial ordering on $R=F[x_{1},\dots,x_{n}]$. A Gröbner basis $\{ g_{1},\dots,g_{m} \}$ for a nonzero ideal $I$ in $R$ is called a reduced Gröbner basis if (a) $LT(g_{i})$ is monic $\forall i$ and (b) no term in $g_{j}$ is divisible by $LT(g_{i}),\ \forall (i,j)$ where $i \neq j$.
Fact: A reduced Gröbner basis is also clearly minimal.
Theorem 27: Fix a monomial ordering on $R=F[x_{1},\dots,x_{n}]$. Then there is a unique reduced Gröbner basis for every nonzero ideal $I$ in $R$.
Corollary 28: Let $I,J$ be ideals in $F[x_{1},\dots,x_{n}]$. Then $I=J\iff$ $I$ and $J$ have the same reduced Gröbner basis with respect to any fixed monomial ordering on $F[x_{1},\dots,x_{n}]$.
Definition: If $I$ is an ideal in $F[x_{1},\dots,x_{n}]$ then $I_{i}=I\cap F[x_{i+1},\dots,x_{n}]=$ set of polynomials in $I$ that do not involve the variables $x_{1},\dots,x_{i}$ is called the $i$th elimination ideal of $I$ with respect to the lexicographic monomial ordering $x_{1}>\dots>x_{n}$.
Proposition 29: (Elimination) Suppose $G=\{ g_{1},\dots,g_{m} \}$ is a Gröbner basis for the nonzero ideal $I$ in $F[x_{1},\dots,x_{n}]$ with respect to the above ordering. Then $G\cap F[x_{i+1},\dots,x_{n}]$ is a Gröbner basis of the $i$th elimination ideal defined above. In particular, $I\cap F[x_{i+1},\dots,x_{n}]=0\iff G\cap F[x_{i+1},\dots,x_{n}]=\emptyset$.
Fact: Let $I,J$ b e ideals in $F[x_{1},\dots,x)_{n}]$ with $I=(f_{1},\dots,f_{s})$ and $J=(h_{1},\dots,h_{t})$. Then $I+J=(f_{1},\dots,f_{s},h_{1},\dots,h_{t})$ and $IJ=(f_{1}h_{1},\dots,f_{i}h_{j},\dots,f_{s}h_{t})$.
Proposition 30: IF $I,J$ are ideals in $F[x_{1},\dots,x_{n}]$ then $tI+(1-t)J$ is an ideal in $F[t,x_{1},\dots,x_{n}]$ and $I\cap J=(tI+(1-t)J)\cap F[x_{1},\dots,x_{n}]$. In particular, $I\cap J$ is the first elimination ideal of $tI+(1-t)J$ with respect to the ordering $t>x_{1}>\dots>x_{n}$.