Chapter 16

Let \(E\) be an \(R\) -algebra. For \(\alpha \in E,\) consider the \(\alpha\) -multiplication map on \(E\), which sends \(\beta \in E\) to \(\alpha \beta \in E\). Show that this map is an \(R\) -linear map.

Show that every ring may be viewed in a unique way as a \(\mathbb{Z}\) algebra, and that subrings are subalgebras, and ring homomorphisms are \(\mathbb{Z}\) -algebra homomorphisms.

Show that the only \(\mathbb{R}\) -algebra homomorphisms from \(\mathbb{C}\) into itself are the identity map and the complex conjugation map.

Let \(F\) be a field of characteristic zero. Show that \(F\) contains an isomorphic copy of \(\mathbb{Q}\).

Show that the field of fractions of \(\mathbb{Z}[i]\) within \(\mathbb{C}\) is \(\mathbb{Q}[i]\). (See Example 7.25 and Exercise \(7.14 .\) )

Let \(\rho: E \rightarrow E^{\prime}\) be an \(F\) -algebra homomorphism, let \(\alpha \in E,\) let \(\phi\) be the minimal polynomial of \(\alpha\) over \(F,\) and let \(\phi^{\prime}\) be the minimal polynomial of \(\rho(\alpha)\) over \(F\). Show that \(\phi^{\prime} \mid \phi,\) and that \(\phi^{\prime}=\phi\) if \(\rho\) is injective.

Let \(\rho: E \rightarrow E^{\prime}\) be an \(F\) -algebra homomorphism, let \(\alpha \in E,\) let \(\phi\) be the minimal polynomial of \(\alpha\) over \(F,\) and let \(\phi^{\prime}\) be the minimal polynomial of \(\rho(\alpha)\) over \(F\). Show that \(\phi^{\prime} \mid \phi,\) and that \(\phi^{\prime}=\phi\) if \(\rho\) is injective.

Show that if the factorization of \(f\) over \(F[X]\) into monic irreducibles is \(f=f_{1}^{e_{1}} \cdots f_{r}^{e_{r}},\) and if \(\alpha=[h]_{f} \in F[X] /(f),\) then the minimal polynomial \(\phi\) of \(\alpha\) over \(F\) is \(\operatorname{lcm}\left(\phi_{1}, \ldots, \phi_{r}\right),\) where each \(\phi_{i}\) is the minimal polynomial of \([h]_{f_{i}^{e_{i}}} \in F[X] /\left(f_{i}^{e_{i}}\right)\) over \(F\)

In the field \(E\) in Example 16.16, find all the elements of degree 2 over \(\mathbb{Z}_{2}\)

Let \(E\) be an extension field of a field \(F,\) and let \(\alpha_{1}, \ldots, \alpha_{n} \in E\) be algebraic over \(F\). Show that the ring \(F\left[\alpha_{1}, \ldots, \alpha_{n}\right]\) (see Example 7.45 ) is in fact a field, and that \(F\left[\alpha_{1}, \ldots, \alpha_{n}\right]\) is a finite (and hence algebraic) extension of \(F\).

Consider the real numbers \(\sqrt{2}\) and \(\sqrt[3]{2}\). Show that \((\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]: \mathbb{Q})=(\mathbb{Q}[\sqrt{2}+\sqrt[3]{2}]: \mathbb{Q})=6\)

Consider the real numbers \(\sqrt{2}\) and \(\sqrt{3}\). Show that \((\mathbb{Q}[\sqrt{2}, \sqrt{3}]: \mathbb{Q})=(\mathbb{Q}[\sqrt{2}+\sqrt{3}]: \mathbb{Q})=4\)

Show that if \(E\) is an algebraic extension of \(K,\) and \(K\) is an algebraic extension of \(F,\) then \(E\) is an algebraic extension of \(F\).

Let \(E\) be an extension of \(F\). Show that the set of all elements of \(E\) that are algebraic over \(F\) is a subfield of \(E\) containing \(F\).

Consider a field \(F\) and its field of rational functions \(F(X) .\) Let \(\alpha \in F(X) \backslash F\). Show that \(X\) is algebraic over \(F(\alpha),\) and that \(\alpha\) is transcendental over \(F\)

Let \(E\) be an extension field of a field \(F\). Suppose \(\alpha \in E\) is transcendental over \(F,\) and that \(E\) is algebraic over \(F(\alpha) .\) Show that for every \(\beta \in E, \beta\) is transcendental over \(F\) if and only if \(E\) is algebraic over \(F(\beta)\)

Prove the "chain rule" for formal derivatives: if \(g, h \in R[X]\) and \(f=g(h) \in R[X],\) then \(\mathbf{D}(f)=\mathbf{D}(g)(h) \cdot \mathbf{D}(h) ;\) more generally, if \(g \in\) \(R\left[X_{1}, \ldots, X_{n}\right],\) and \(h_{1}, \ldots, h_{n} \in R[X],\) and \(f=g\left(h_{1}, \ldots, h_{n}\right) \in R[X],\) then $$ \mathbf{D}_{X}(f)=\sum_{i=1}^{n} \mathbf{D}_{X_{i}}(g)\left(h_{1}, \ldots, h_{n}\right) \mathbf{D}_{X}\left(h_{i}\right) $$

Let \(g \in R[X],\) and let \(x \in R\) be a root of \(g .\) Show that \(x\) is a multiple root of \(g\) if and only if \(x\) is also a root of \(\mathbf{D}(g)\) (see Exercise 7.18 ).

Let \(g \in R[X]\) with \(\operatorname{deg}(g)=k \geq 0,\) and let \(x \in R .\) Show that if we evaluate \(g\) at \(X+x,\) writing $$ g(X+x)=\sum_{i=0}^{k} b_{i} X^{i} $$ with \(b_{0}, \ldots, b_{k} \in R,\) then we have $$ i ! \cdot b_{i}=\left(\mathbf{D}^{i}(g)\right)(x) \text { for } i=0, \ldots, k $$

Suppose \(p\) is a prime, \(g \in \mathbb{Z}[X],\) and \(x \in \mathbb{Z},\) such that \(g(x) \equiv 0(\bmod p)\) and \(\mathbf{D}(g)(x) \not \equiv 0(\bmod p)\). Show that for every positive integer \(e,\) there exists an integer \(\hat{x}\) such that \(g(\hat{x}) \equiv 0\left(\bmod p^{e}\right),\) and give an efficient procedure to compute such an \(\hat{x}\), given \(p, g, x,\) and \(e\). Hint: mimic the "lifting" procedure discussed in \(\S 12.5 .2\)

Let \(F\) be a field. Show that every non-zero ideal of \(F \llbracket X \rrbracket\) is of the form \(\left(X^{m}\right)\) for some uniquely determined integer \(m \geq 0\).

Let \(F\) be a field. Show that \(F((X))\) is the field of fractions of \(F \llbracket X \rrbracket ;\) that is, there is no subfield \(E \subsetneq F((X))\) that contains \(F \llbracket X \rrbracket\).

Write down the rule for determining the multiplicative inverse of an element of \(R\left(\left(X^{-1}\right)\right)\) whose leading coefficient is a unit in \(R\).

Let \(F\) be a field of characteristic other than \(2 .\) Show that a non-zero \(g \in F\left(\left(X^{-1}\right)\right)\) has a square-root in \(F\left(\left(X^{-1}\right)\right)\) if and only if \(\operatorname{deg}(g)\) is even and \(\operatorname{lc}(g)\) has a square-root in \(F\).

Let \(R\) be a ring, and let \(a \in R\). Show that the multiplicative inverse of \(X-a\) in \(R\left(\left(X^{-1}\right)\right)\) is \(\sum_{j=1}^{\infty} a^{j-1} X^{-j}\)

Let \(R\) be an arbitrary ring, let \(a_{1}, \ldots, a_{\ell} \in R,\) and let $$ f:=\left(X-a_{1}\right)\left(X-a_{2}\right) \cdots\left(X-a_{\ell}\right) \in R[X] $$ For \(j \geq 0,\) define the "power sum" $$ s_{j}:=\sum_{i=1}^{\ell} a_{i}^{j} $$ Show that in the ring \(R\left(\left(X^{-1}\right)\right)\), we have $$ \frac{\mathbf{D}(f)}{f}=\sum_{i=1}^{\ell} \frac{1}{\left(X-a_{i}\right)}=\sum_{j=1}^{\infty} s_{j-1} X^{-j} $$ where \(\mathbf{D}(f)\) is the formal derivative of \(f\).

(a) Show that the "is associate to" relation is an equivalence relation. (b) Consider an equivalence class \(C\) induced by the "is associate to" relation. Show that if \(C\) contains an irreducible element, then all elements of \(C\) are irreducible. (c) Suppose that for every equivalence class \(C\) that contains irreducibles, we choose one element of \(C,\) and call it a distinguished irreducible. Show that \(D\) is a UFD if and only if every non-zero element of \(D\) can be expressed as \(u p_{1}^{e_{1}} \cdots p_{r}^{e_{r}},\) where \(u\) is a unit, \(p_{1}, \ldots, p_{r}\) are distinguished irreducibles, and this expression is unique up to a reordering of the \(p_{i}\) 's.

Show that the ring \(\mathbb{Z}[\sqrt{-5}]\) is not a UFD.

Let \(D\) be a UFD and \(F\) its field of fractions. Show that (a) every element \(x \in F\) can be expressed as \(x=a / b,\) where \(a, b \in D\) are relatively prime, and (b) that if \(x=a / b\) for \(a, b \in D\) relatively prime, then for any other \(a^{\prime}, b^{\prime} \in D\) with \(x=a^{\prime} / b^{\prime}\), we have \(a^{\prime}=c a\) and \(b^{\prime}=c b\) for some \(c \in D\).

Let \(D\) be a UFD and let \(p \in D\) be irreducible. Show that there is no prime ideal \(Q\) of \(D\) with \(\left\\{0_{D}\right\\} \subsetneq Q \subsetneq p D\) (see Exercise 7.38).

Consider the polynomial $$ X^{3}-1=(X-1)\left(X^{2}+X+1\right) $$ Over \(\mathbb{C},\) the roots of \(X^{3}-1\) are \(1,(-1 \pm \sqrt{-3}) / 2 .\) Let \(\omega:=(-1+\sqrt{-3}) / 2,\) and note that \(\omega^{2}=-1-\omega=(-1-\sqrt{-3}) / 2,\) and \(\omega^{3}=1\). (a) Show that the ring \(\mathbb{Z}[\omega]\) consists of all elements of the form \(a+b \omega,\) where \(a, b \in \mathbb{Z},\) and is an integral domain. This ring is called the ring of Eisenstein integers. (b) Show that the only units in \(\mathbb{Z}[\omega]\) are \(\pm 1, \pm \omega,\) and \(\pm \omega^{2}\). (c) Show that \(\mathbb{Z}[\omega]\) is a Euclidean domain.

Show that in a PID, all non-zero prime ideals are maximal (see Exercise 7.38 ). Recall that for a complex number \(\alpha=a+b i\), with \(a, b \in \mathbb{R},\) the norm of \(\alpha\) was defined as \(N(\alpha)=\alpha \bar{\alpha}=a^{2}+b^{2}\) (see Example 7.5). There are other measures of the "size" of a complex number that are useful. The absolute value of \(\alpha\) is defined as \(|\alpha|:=\sqrt{N(\alpha)}=\sqrt{a^{2}+b^{2}}\). The max norm of \(\alpha\) is defined as \(M(\alpha):=\max \\{|a|,|b|\\}\)

Let \(\alpha, \beta \in \mathbb{C}\). Prove the following statements: (a) \(|\alpha \beta|=|\alpha||\beta| ;\) 16.9 Unique factorization domains (*) (b) \(|\alpha+\beta| \leq|\alpha|+|\beta| ;\) (c) \(N(\alpha+\beta) \leq 2(N(\alpha)+N(\beta)) ;\) (d) \(M(\alpha) \leq|\alpha| \leq \sqrt{2} M(\alpha)\)

Let \(\pi:=1+i \in \mathbb{Z}[i]\) (a) Show that \(2=\pi \bar{\pi}=-i \pi^{2},\) that \(N(\pi)=2,\) and that \(\pi\) is irreducible in \(\mathbb{Z}[i]\). \(458 \quad\) More rings (b) Let \(\alpha \in \mathbb{Z}[i],\) with \(\alpha=a+b i\) for \(a, b \in \mathbb{Z} .\) Show that \(\pi \mid \alpha\) if and only if \(a-b\) is even, in which case $$ \frac{\alpha}{\pi}=\frac{a+b}{2}+\frac{b-a}{2} i $$ (c) Show that for all \(\alpha \in \mathbb{Z}[i],\) we have \(\alpha \equiv 0(\bmod \pi)\) or \(\alpha \equiv 1(\bmod \pi)\). (d) Show that the quotient ring \(\mathbb{Z}[i] / \pi \mathbb{Z}[i]\) is isomorphic to the ring \(\mathbb{Z}_{2}\). (e) Show that for all \(\alpha \in \mathbb{Z}[i]\) with \(\alpha \equiv 1(\bmod \pi),\) there exists a unique \(\varepsilon \in\\{\pm 1, \pm i\\}\) such that \(\alpha \equiv \varepsilon(\bmod 2 \pi)\) (f) Show that for all \(\alpha, \beta \in \mathbb{Z}[i]\) with \(\alpha \equiv \beta \equiv 1(\bmod \pi),\) there exists a unique \(\varepsilon \in\\{\pm 1, \pm i\\}\) such that \(\alpha \equiv \varepsilon \beta(\bmod 2 \pi)\)

We now present a "( \(1+i\) )-ary gcd algorithm" for Gaussian integers. Let \(\pi:=1+i \in \mathbb{Z}[i] .\) The algorithm takes non-zero \(\alpha, \beta \in \mathbb{Z}[i]\) as input, and runs as follows: $$ \begin{array}{l} \rho \leftarrow \alpha, \rho^{\prime} \leftarrow \beta, e \leftarrow 0 \\ \text { while } \pi \mid \rho \text { and } \pi \mid \rho^{\prime} \text { do } \rho \leftarrow \rho / \pi, \rho^{\prime} \leftarrow \rho^{\prime} / \pi, e \leftarrow e+1 \end{array} $$ repeat while \(\pi \mid \rho\) do \(\rho \leftarrow \rho / \pi\) while \(\pi \mid \rho^{\prime}\) do \(\rho^{\prime} \leftarrow \rho^{\prime} / \pi\) if \(M\left(\rho^{\prime}\right)

In Exercise 16.41, we saw that 2 factors as \(-i(1+i)^{2}\) in \(\mathbb{Z}[i]\), 16.9 Unique factorization domains (*) where \(1+i\) is irreducible. This exercise examines the factorization in \(\mathbb{Z}[i]\) of prime numbers \(p>2\). Show that: (a) for every irreducible \(\pi \in \mathbb{Z}[i],\) there exists a unique prime number \(p\) such that \(\pi\) divides \(p\); (b) for all prime numbers \(p \equiv 1(\bmod 4),\) we have \(p=\pi \bar{\pi},\) where \(\pi \in \mathbb{Z}[i]\) is irreducible, and the complex conjugate \(\bar{\pi}\) of \(\pi\) is also irreducible and not associate to \(\pi\); (c) all prime numbers \(p \equiv 3(\bmod 4)\) are irreducible in \(\mathbb{Z}[i]\).

Let \(D\) be a UFD, let \(p\) be an irreducible element of \(D\), and consider the natural map that sends \(a \in D\) to \(\bar{a}:=[a]_{p} \in D / p D,\) which we extend coefficient-wise to a ring homomorphism from \(D[X]\) to \((D / p D)[X]\) (see Example 7.46 ). Show that if \(f \in D[X]\) is a primitive polynomial such that \(p \nmid \operatorname{lc}(f)\) and \(\bar{f} \in(D / p D)[X]\) is irreducible, then \(f\) is irreducible.

Let \(a\) be a non-zero, square-free integer, with \(a \notin\\{\pm 1\\},\) and let \(n\) be a positive integer. Show that the polynomial \(X^{n}-a\) is irreducible in \(\mathbb{Q}[X]\)

Show that the polynomial \(X^{4}+1\) is irreducible in \(\mathbb{Q}[X]\).

Let \(F\) be a field, and consider the ring of bivariate polynomials \(F[X, Y]\). Show that in this ring, the polynomial \(X^{2}+Y^{2}-1\) is irreducible, provided \(F\) does not have characteristic 2 . What happens if \(F\) has characteristic \(2 ?\)

Design and analyze an efficient algorithm for the following problem. The input is a pair of polynomials \(g, h \in \mathbb{Z}[X],\) along with their greatest common divisor \(d\) in the ring \(\mathbb{Q}[X] .\) The output is the greatest common divisor of \(g\) and \(h\) in the ring \(\mathbb{Z}[X]\)

Let \(g, h \in \mathbb{Z}[X]\) be non-zero polynomials with \(d:=\operatorname{gcd}(g, h) \in\) \(\mathbb{Z}[X]\). Show that for every prime \(p\) not dividing \(\operatorname{lc}(g) \operatorname{lc}(h),\) we have \(\bar{d} \mid \operatorname{gcd}(\bar{g}, \bar{h})\), and except for finitely many primes \(p,\) we have \(\bar{d}=\operatorname{gcd}(\bar{g}, \bar{h}) .\) Here, \(\bar{d}, \bar{g},\) and \(\bar{h}\) denote the images of \(d, g,\) and \(h\) in \(\mathbb{Z}_{p}[X]\) under the coefficient-wise extension of the natural map from \(\mathbb{Z}\) to \(\mathbb{Z}_{p}\) (see Example 7.47 ).