2.2 PID’s, UFD’s and polynomial rings

KNOW: how to compute the gcd and the lcm of two elements $a$ and $b$ in a UFD $R$ and that the gcd can in general not be written as an $R$-linear combination of $a$ and $b$ (as opposed to the case where $R$ is a PID); standard examples of non-principal ideals in $\mathbb{Z}[x]$ and $K[x, y]$ for a field $K$; the difference between irreducible and prime elements (for instance in a ring like $\mathbb{Z}[\sqrt{-5}]$ ) and that these two notions coincide in a UFD; the fact that prime elements generate prime ideals which are minimal among the nonzero prime ideals if $R$ is a UFD; the fact that in any integral domain irreducible elements generate principal ideals which are maximal among the proper principal ideals.

Know Hilbert’s Basis Theorem: If $R$ is Noetherian, then also the polynomial ring $R[x]$ is Noetherian, i.e. all its ideals are finitely generated.

Know Gauss’ Lemma and its variants: If $R$ is a UFD, $f, g \in R[x]$ with $f$ primitive and $f \mid g$ in $F[x]$, where $F$ is the field of fractions of $R$, then $f \mid g$ in $R[x]$; primitive polynomials of $R[x]$ are irreducible iff they are prime in $R[x]$ iff they are irreducible (or prime) in $F[x]$;
Consequence: $R[x]$ is also a UFD, in particular $\mathbb{Z}\left[x_{1}, \ldots, x_{n}\right]$ and $K\left[x_{1}, \ldots, x_{n}\right]$ are UFD’s for any $n \in \mathbb{N}$ and any field $K$.

Know EISENSTEIN’S CRITERION: If $f \in R[x]$ is a primitive polynomial ( $R$ a UFD) and $p$ a prime dividing all coefficients of $f$ but the leading one and $p^{2}$ not dividing the constant term, then $f$ is irreducible in $R[x]$ and hence in $F[x]$.

Avoid the following mistakes: If $R$ is a subring of $S$ and $p$ is a prime element in $R, p$ need not be a prime element in $S$ (even if $S$ is a UFD and $p$ is not a unit in $S$ ); if you want to deduce from an equation $a=b c$ in a domain $R$ that $a$ is not irreducible, don’t forget to check that $a$ and $b$ are no units in $R$; likewise, if you have different irreducible elements $a, b, c, d$ in $R$ and you want to deduce from an equation $a b=c d$ that they are not prime in $R$, don’t forget to check that they are not associate; keep in mind that the primitivity of $f$ is an important assumption in Gauss’ Lemma; don’t apply Eisenstein’s criterion to $\mathbb{F}_{q}[x]$ ( $\mathbb{F}_{q}$ is not the field of fractions of a proper subdomain, and it doesn’t have any prime elements).

Related Problems

(Unless otherwise stated, all rings $R$ are commutative with unit $1 ; F$ denotes a field.)

1. (April $77 \# 4$ ) (a) If $M$ is a maximal ideal, SHOW $R / M$ is a field.
   (b) If $R$ is a Euclidean domain, show every ideal generated by an irreducible element is maximal.

2. (Jan 79 #2; May 91 #2a) (a) SHOW that any Euclidean domain is a PID.
   (b) Show that any two nonzero elements have a g.c.d. in $R$ if $R$ is a Euclidean domain.
   (c)Find (systematically) $m, n \in \mathbb{Z}$ so that $421 m+1664 n=1$.

3. (Sep 79 #7b) Given an example of a prime ideal in $R$ which is not maximal; is there an example where $R$ is a UFD?

4. (Sep 80 #2) (a) Give 3 examples of PIDs.
   (b) Show that the homomorphic image of a PIR (Principal Ideal Ring: all ideals are principal, but perhaps not a domain) is again a PIR.
   (c) Give an example of a PID with a homomorphic image which is not a PID.

5. (Jan 81 #4) SHOW $F[x]$ is a Euclidean domain, but $F[x, y]$ is not.

6. (Sep $82 \# 2$ ) If $a_{1}, \ldots, a_{n}$ in a PID $R$ have $\operatorname{gcd} d$, show that there exists an invertible $n \times n$ matrix $Q$ of determinant 1 over $R$ with $Q\left[a_{1}, \ldots, a_{n}\right]^{T}=[d, 0,0 \ldots 0]^{T}$. (This is false if $R$ is merely a UFD).

7. (Mar $83 \# 5$ ) If $D$ is a UFD whose units together with 0 form a proper subring $U$, show $D$ has infinitely many (nonassociate) primes. Give an example of such a $D$.

8. (Sep $83 \# 7$ ) If $R \subseteq S$ are PIDs with $d=\operatorname{gcd}_{R}(a, b)$, show $d=\operatorname{gcd}_{S}(a, b)$ too.

9. (Feb $84 \# 5$ ) Factor $x^{3}-y^{3}$ into irreducible factors in $\mathbb{Q}[x, y]$.

10. (1985 #3a) Show $x^{5}-6 x^{3}+12 x^{2}+21 x-3$ is irreducible in $\mathbb{Q}[x]$.

11. (Jan $87 \# 5$ ) If $R$ is an integral domain, what are necessary and sufficient conditions that $R[x]$ be: (0) a domain, (1) a PID, (2) a UFD, (3) noetherian.

12. (Fall $87 \# 2$ ) Prove or disprove: $R$ PID $\Longrightarrow R[x]$ is PID.

13. (Aug $88 \# 1$ ) If $F$ is a field, PROVE $F[x]$ is a PID.

14. (Jan 89 #6) Prove or disprove: Every UFD is a PID.

15. (Aug 89 #4) Show $R=\{f(x) \in \mathbb{Z}[x] \mid$ the coefficient of $x$ in $f(x)$ is even $\}$ is a subring. Show that 2 and $2 x$ have a g.c.d. in $R$, but not a l.c.m.

16. (May $1990 \# 4$ ) If $P$ is a nonzero prime ideal in a UFD, show $P$ is minimal among nonzero prime ideals iff $P$ is a principal ideal.

17. (May 91 #2b) Prove or disprove: $R$ Euclidean domain $\Longrightarrow R[x]$ Euclidean domain and/or any two nonzero elements of $R[x]$ have a g.c.d. in $R[x]$.

18. (Jan 92 #2) Show that a polynomial of degree $n$ over $F$ has at most $n$ roots in $F$.

19. (May 92 #6) Show that $f(x, y)=x+x^{3} y+y^{8}+x^{7} y^{5}+x^{2} y^{4}$ is irreducible over the rational field.

20. (Jan $94 \# 3$ ) Prove that $y^{3}+x^{2} y^{2}+x^{3} y+x$ is irreducible in $\mathbb{Z}[x, y]$.

21. (Aug 94 #3) Describe which polynomials in $\mathbb{R}[x]$ belong to the subring $\mathbb{R}\left[x^{2}, x^{3}\right], \mathbb{R}$ the field of real numbers.

22. (Jan 95 #6) Prove that any proper homomorphic image of a PID that remains an integral domain must actually be a field.

23. (Aug $95 \# 4$ ) Show that $\mathbb{Z}[\sqrt{10}]=\mathbb{Z}+\mathbb{Z} \sqrt{10}$ is not a UFD. (Hint: show that a) $n^{2}-10 m^{2} \neq \pm 2,3$ for all $n, m \in \mathbb{Z}$, b) 2,3 , and $4 \pm \sqrt{10}$ are irreducible in $\mathbb{Z}[\sqrt{10}]$.)

24. (Aug $95 \# 5$ ) Factor $x^{9}-x$ in $\mathbb{F}_{3}[x]$ into irreducible factors ( $\mathbb{F}_{3}$ the Galois field of three elements).

25. (Aug $96 \# 4)$ In $\mathbb{Q}[x]$, let $f(x)=x^{m_{1}}+\ldots+x^{m_{k}}$ where $m_{i} \equiv i-1(\bmod k)$. Show that $f(x)$ is divisible by $x^{k-1}+x^{k-2}+\ldots+1$.

26. (Aug $96 \# 5$ ) Let $R$ be a PID. An ideal $P$ of $R$ is called primary if whenever $a b \in P$ and $a \notin P$ then $b^{n} \in P$ for some $n$ (depending on $b$ ). Show that $P$ is primary iff either $P=0$ or $P=\left(p^{m}\right)$ for some prime $p \in R$ and some exponent $m$.

27. (Jan 97 #4) Give counterexamples for each of the following statements, with details. Then correct each statement by modifying the underlined part. (a) If $R$ is a commutative ring, then a polynomial in $R[x]$ of degree $n$ has at most $n$ roots in $R$. (b) If $R$ is a division ring, then a polynomial in $R[x]$ of degree $n$ has at most $n$ roots in $R$. (c) If $R$ is a unique factorization domain, then the greatest common divisor $d$ of two members $a, b$ of $R$ can be written as $d=a x+b y$ for some $x$ and $y$ in $R$.

28. (Jan $98 \# 3$ ) Let $R$ be a PID. If $a, b$ are two nonzero elements in $R$, show that they have a l.c.m. (an element $m \in R$ such that (1) $a, b \mid m$, (2) if $a, b \mid x$ then $m \mid x$.)

29. (August 98 #5) The Krull dimension of a commutative ring $R$ is the longest chain of prime ideals properly contained in $R$, i.e. the largest integer $n$ such that there exists a chain $P_{0}<P_{1}<\ldots<P_{n}<R$ ( $P_{0}=0$ allowed if prime) of prime ideals $P_{i}$ in $R$. If $R$ is a PID, find its Krull dimension.

30. (Jan $00 \# 6$ ) Given a finite field $K$, show there exists a polynomial $f(x, y) \in K[x, y]$ (in which both variables actually appear) for which the equation $f(x, y)=0$ has no solutions in $K \times K$.

31. (Aug $01 \# 4$ ) If $x$ is an element of a PID $R$, show that the left $R$-module $R / R x$ is irreducible as a module if and only if the element $x$ is irreducible as an element.

32. (Aug 02 #1) (a) Find all nonconstant polynomials $p(x) \in \mathbb{C}[x]$ satisfying $p(p(p((x))=p(x)$. (b) Find all rational functions $\varphi(x) \in \mathbb{C}(x)$ satisfying $\varphi(2 x)=\varphi(x)$.

33. (Aug $02 \# 6$ ) Let $R=\mathbb{C}[x, y] / \mathfrak{a}$ where $\mathbb{C}[x, y]$ is the ring of polynomials in variables $x$ and $y$, and $\mathfrak{a}$ is the principal ideal of $\mathbb{C}[x, y]$ generated by $p(x, y)=y^{2}-x^{3}$. Show that the ideal of $R$ generated by the image of $x$ and $y$ is not principal.

34. (May $03 \# 7$ ) Let $f(x) \in \mathbb{R}[x]$ be a nonzero polynomial. Show that there exists a number $r \in \mathbb{R}$ such that the polynomials $f(x)$ and $f(x+r)$ are relatively prime.

35. (Aug 03, #2) Let $D$ be a PID with $F$ as its field of fractions. Show that every element $x \in F$ can be written as a sum of primary fractions (i.e. with denominators powers of primes):

$$x=\sum_{i=1}^{n} \frac{a_{i}}{p_{i}^{e_{i}}}$$

for some $a_{1}, \ldots, a_{n} \in D$ and non-associate primes $p_{1}, \ldots, p_{n} \in D$.
36. (Jan 04 #4) Show that a principal ideal in $\mathbb{Z}[x]$ can never be maximal.
37. (Aug 04 #3) Consider the ring $R=\mathbb{Z}[\sqrt{-7}]=\{m+n \sqrt{-7} \mid m, n \in \mathbb{Z}\}$.
(a) Is $R$ a UFD? Give arguments for your answer.
(b) Exhibit an ideal $I$ in $R$ which is not principal. Show that your $I$ is not principal.
38. (Jan $05 \# 5$ ) Prove that the ideal $\left(x^{2}+2, x^{2}+7\right)$ is maximal in $\mathbb{Z}[x]$.
39. (Jan 06 #8) For an element $a \in \mathbb{Q}$, consider the homomorphism $\varphi$ from the polynomial ring $\mathbb{Q}[x]$ to the ring $M_{3}(\mathbb{Q})$ of $3 \times 3$ matrices, given by evaluation $f(x) \mapsto f(A)$ for

$$A=\left(\begin{array}{ccc} a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & a \end{array}\right)$$

Show that the kernel of $\varphi$ is the set of all polynomials $f(x)$ with $f(a)=f^{\prime}(a)=f^{\prime \prime}(a)=0$. Is this a principal ideal?
40. (Aug 06 #4) Decide whether

$$x y^{2}+x^{2} y+2 x y+y+x+1$$

is irreducible in $\mathbb{Q}[x, y]$.
41. (Aug $06 \# 7$ ) Determine the number of monic irreducible polynomials of degree 2 in $\mathbb{F}_{7}[x]$.
42. (Jan 08 #4) (a) If $r \in \mathbb{Q}$ is a rational root of a monic integral polynomial

$$p(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \in \mathbb{Z}[x]$$

show that $r \in \mathbb{Z}$ is integral.
(b) Factor $y^{5} x+y^{3} x^{2}+y+x^{3}$ into irreducible factors in $\mathbb{Z}[x, y]$, explaining why each is irreducible.
43. (Jan 08 #5) Let $R$ be a unital commutative ring, and consider 5 possible properties such a ring might have: it is $\left(\mathcal{P}_{1}\right)$ a domain, $\left(\mathcal{P}_{2}\right)$ a PID, $\left(\mathcal{P}_{3}\right)$ Euclidean, $\left(\mathcal{P}_{4}\right)$ noetherian, $\left(\mathcal{P}_{5}\right)$ a UFD.
(a) For which $n$ is it true that $R[x]$ always inherits property $\left(\mathcal{P}_{n}\right)$ from $R$ (if $R$ has $\left(\mathcal{P}_{n}\right)$, so must $R[x])$ ? Circle the $n^{\prime} s$ for which this holds, and explain your answer or give a counterexample.

$$n=\begin{array}{|lllll|} \hline 1 & 2 & 3 & 4 & 5 \\ \hline \end{array}$$

(b) For which $n$ is it true that $R$ inherits property $\left(\mathcal{P}_{n}\right)$ from $R[x]$ ? Circle the $n^{\prime} s$ for which this holds, and explain your answer or give a counterexample.

$$n=\begin{array}{|lllll|} \hline 1 & 2 & 3 & 4 & 5 \\ \hline \end{array}$$

(c) For which $n \neq 5$ is it true that a quotient $R / P$ of $R$ by a prime ideal ( $P \triangleleft R, P \neq R$ ) inherits property $\left(\mathcal{P}_{n}\right)$ from $R$ ? Circle the $n^{\prime} s$ for which this holds, and explain your answer or give a counterexample.

$$n=\begin{array}{|llll|} \hline 1 & 2 & 3 & 4 \\ \hline \end{array}$$

44. (Jan 08 #6) Let $R=\mathbb{Z}+2 \mathbb{Z}[x]=\mathbb{Z} 1+\sum_{n=0}^{\infty} 2 \mathbb{Z} x^{n}$.
    (a) Show that $R$ is not a UFD by finding an irreducible element that is not prime.
    (b) Show that $R$ is not noetherian by showing that the ideal $2 \mathbb{Z}[x]$ is not finitely generated: $2 \mathbb{Z}[x] \neq \sum_{i=1}^{n} R f_{i}(x)$ for any $f_{i}(x) \in 2 \mathbb{Z}[x]$.

45. (Jan $09 \# 3$ ) Let $F$ be a field, and let $R$ be the subring of $F[x]$ consisting of all polynomials with zero coefficient of $x$, that is,

$$R=\left\{a_{0}+a_{2} x^{2}+\ldots+a_{n} x^{n}: a_{i} \in F\right\}$$

(a) Prove that the elements $x^{2}$ and $x^{3}$ are irreducible but not prime in $R$.
(b) Is $R$ a principal ideal domain? Prove your answer.
(c) Prove that $R$ is Noetherian.
46. (Aug $09 \# 3$ )(a) Let $R$ be a principal ideal domain and $I \subset R$ a proper nonzero ideal. Prove that if the quotient ring $R / I$ is a domain then it must be a field.
(b) Does the assertion of (a) remain true if $R$ is only assumed to be a unique factorization domain? Prove or give a counterexample.
47. (Aug $09 \# 4$ ) Let $F$ be a field and $R=F[x, y]$ the ring of polynomials in two (commuting) variables $x$ and $y$. Let $I=x R$ be the principal ideal of $R$ generated by $x$ and $S=F+I= \{f+i: f \in F, i \in I\}$. Observe that S is a subring of $R$ and $I$ is an ideal of $S$ (you need to justify these facts).
(a) Prove that $I$ is not finitely generated as an ideal of $S$.

Hint: Assume that $I$ is finitely generated as an ideal of $S$ and reach a contradiction by showing that there must exist a natural number $m$ such that any polynomial $p(x, y) \in I$ contains no monomials of the form $x y^{n}$, with $n>m$.
(b) Prove that $S$ is not finitely generated as a ring.

Hint: It is possible to answer (b) using (a) without doing any computations.
48. (Aug $10 \# 3$ ) Let $\mathbb{R}$ denote the real numbers. The purpose of this problem is to show that the ring $A=\mathbb{R}[x, y] /\left(x^{2}+y^{2}-1\right)$ is not a UFD. For an element $f \in \mathbb{R}[x, y]$ we denote its image in $A$ by $[f]$.
(a) Show that every element of $A$ can be uniquely expressed in the form $[f(x)+g(x) y]$ where $f(x), g(x) \in \mathbb{R}[x]$.
(b) Show that $A$ has an automorphism $\varphi$ of order 2 such that $\varphi([f(x)])=[f(x)]$ for each $f(x) \in \mathbb{R}[x]$ and $\varphi([y])=-[y]$.
(c) Use (a) and (b) to construct a function $N: A \rightarrow \mathbb{R}[x]$ such that $N(u v)=N(u) N(v)$ for all $u, v \in A$.
(d) Use the function $N$ from (c) to show that $[x]$ is an irreducible element of $A$ and that the only invertible elements of $A$ are (images of) nonzero constant polynomials.
Hint: It is essential that you are working over $\mathbb{R}$, not over $\mathbb{C}$.
(e) Now show that $A$ is not a UFD.
49. (Aug 10 #4) Recall that a ring $S$ is called graded if $S=\oplus_{n=0}^{\infty} S_{n}$ where each $S_{n}$ is an additive subgroup and $S_{n} \cdot S_{m} \subseteq S_{n+m}$ for all $n, m$. An element $s \in S$ is called homogeneous if $s \in S_{n}$ for some $n$. An ideal $I$ of $S$ is called a graded ideal if $I=\oplus_{n=0}^{\infty}\left(I \cap S_{n}\right)$.
(a) Let $S$ be a graded ring and $I$ and ideal of $S$. Prove that $I$ is a graded ideal if and only if $I$ is generated (as an ideal) by a set of homogeneous elements.
(b) Let $R$ be a commutative ring with 1 and $S=R[x]$. Then $S$ is a naturally graded ring where $S_{n}=\left\{r x^{n}: r \in R\right\}$. Assume that there exists $k \in \mathbb{N}$ such that every ideal of $R$ can be generated by at most $k$ elements.
Let $I$ be a graded ideal of $S$. Prove that $I$ can be written as $I=J \oplus M$ where $J$ is an ideal of $S$ generated by at most $k$ elements and $M$ is a finitely generated $R$-module.
Hint: Adapt the proof of Hilbert’s basis theorem.
50. (Aug 11 # 4a) Find all the irreducible polynomials of degree 4 over the finite field $\mathbb{F}_{2}$.
51. (Jan $12 \# 7$ ) If $q$ is a prime power, denote by $\mathbb{F}_{q}$ a finite field of order $q$.
(a) Find a monic irreducible polynomial of degree 3 over $\mathbb{F}_{5}$ and use it to construct a field of order 125 . Justify your answer.
(b) Find all $q$ for which the polynomial $p(x)=x^{2}+x+1$ is irreducible in $\mathbb{F}_{q}[x]$. Hint: What can you say about roots of $p(x)$, and what do you know about the multiplicative group $\mathbb{F}_{q}^{\times}$?
52. (Aug $12 \# 3$ ) Let $k$ be a field and $R=k[x, y] /\left(x^{5}-y^{2}\right)$.
(a) Prove that $R$ is isomorphic to the subring $k\left[t^{2}, t^{5}\right]$ of $k[t]$ (the polynomials in one variable over $k)$.
(b) Prove that $R$ is not isomorphic to $k[t]$ (as a ring).
53. (Jan $13 \# 3$ ) Let $F$ be a field, $d$ a positive integer, and $f_{1}, f_{2}, \ldots \in F\left[x_{1}, \ldots, x_{d}\right]$ an infinite sequence of polynomials in $F\left[x_{1}, \ldots, x_{d}\right]$. Given a positive integer $n$, let $S_{n}$ be the set of all $d$-tuples $\left(a_{1}, \ldots, a_{d}\right) \in F^{d}$ satisfying the following system of equations:

$$f_{i}\left(a_{1}, \ldots, a_{d}\right)=0 \text { for each } 1 \leq i \leq n-1 \text { and } f_{n}\left(a_{1}, \ldots, a_{d}\right)=1$$

Prove that there exists an integer $N$ such that the set $S_{n}$ is empty for all $n \geq N$. Hint: Noetherian rings.
54. (Aug $15 \# 2$ ) Let $r_{1}, r_{2}, r_{3}$ be the roots of the cubic polynomial $X^{3}+10 X^{2}-5 X+4$. Find the cubic polynomial with rational coefficients whose roots are $r_{1}^{2}, r_{2}^{2}, r_{3}^{2}$.
55. (Jan $16 \# 1$ ) Let $K$ be a field (possibly finite). Prove that the polynomial ring $K[X]$ has infinitely many maximal ideals.
56. (Aug $17 \# 5$ ) Let $R=\mathbb{Z}[x] /\left(x^{3}+x^{2}+1\right)$ be the quotient of the polynomial ring $\mathbb{Z}[x]$ modulo the principal ideal $\left(x^{3}+x^{2}+1\right)$.
(a) Is $R$ an integral domain?
(b) Which of the principal ideals (2), (3), (5) of $R$ are prime ideals? And which of them are maximal?
57. (Jan $18 \# 2$ ) Consider the ring $R=\mathbb{Z}[\sqrt{-11}]=\{m+n \sqrt{-11} \mid m, n \in \mathbb{Z}\}$.
(a) Is $R$ a UFD? Give arguments for your answer.
(b) Exhibit an ideal $I$ in $R$ which is not principal. Show that your $I$ is not principal.
58. (Jan 18 #3) Decide in each of the following three cases whether the given polynomial is irreducible. Include arguments.
(a) $x^{2}-2 i$ in $\mathbb{Z}[i][x]$;
(b) $x^{3}-49 x^{2}+(3+\sqrt{2}) x+7$ in $\mathbb{Z}[\sqrt{2}][x]$;
(c) $x^{2}+x y+y^{2}$ in $\mathbb{C}[x, y]$.
59. (Jan $19 \# 3$ ) Let $R=\mathbb{Z}[x]$. Find the number of maximal ideals of $R$ which contain $x^{2}+1$ and 15, and find explicit generators for each such ideal.
Hint: Reduce to a question about Gaussian integers.
60. (Jan $21 \# 8$ ) Let $p \neq 3$ be a prime and $R=\mathbb{F}_{p}[x] /\left(x^{3}-1\right)$. Describe the multiplicative group $R^{\times}$as a direct product of cyclic groups.
Note: Your answer can (and should) involve cases, but should be expressed explicitly in terms of $p$.
61. (Aug 21 #4) Let $R$ be a (commutative) UFD (with 1 ), let $f \in R$, and assume that $f$ is non-unit and nonzero. Let $R_{f}=R\left[\frac{1}{f}\right]$ (you can think of $R$ as the ring of fractions of $R$ with the set of denominators $D=\left\{1, f, f^{2}, \ldots\right\}$ or as the subring of the field of fractions $\operatorname{Frac}(R)$ generated by $R$ and $\frac{1}{f}$ ). Prove that

$$R_{f}^{\times} \cong R^{\times} \times \mathbb{Z}^{m} \text { for some } m \in \mathbb{N}$$

(Here $S^{\times}$is the multiplicative group of $S$ ). Give a detailed argument.
62. (Jan $22 \# 3$ ) Let $f(x, y), g(x, y) \in \mathbb{C}[x, y]$ be two polynomials that do not have a (non-constant) common factor.

(1) Show that $f$ and $g$ are relatively prime as elements of $\mathbb{C}(x)[y]$ and $\mathbb{C}(y)[x]$ (where $\mathbb{C}(x)$ and $\mathbb{C}(y)$ are the fields of rational functions, i.e. the fraction fields of $\mathbb{C}[x]$ and $\mathbb{C}[y]$, respectively). Give a detailed argument.

(2) Show that the system of polynomial equations $f(x, y)=0$ and $g(x, y)=0$ has finitely many solutions in $\mathbb{C}^{2}$. Hint: Use (1) and the fact that $\mathbb{C}(x)[y]$ and $\mathbb{C}(y)[x]$ are PIDs (make sure to explain why the latter is true).