Chapter 16

(a) Use the definition of a field to show that \(\mathbb{Q}(\sqrt{2})\) is a field. (b) Use the definition of vector space to show that \(\mathbb{Q}(\sqrt{2})\) is a vector space over \(\mathbb{Q}\). (c) Prove that \(\\{1, \sqrt{2}\\}\) is a basis for the vector space \(\mathbb{Q}(\sqrt{2})\) over \(\mathbb{Q}\), and, therefore, the dimension of \(\mathbb{Q}(\sqrt{2})\) over \(\mathbb{Q}\) is 2 .

Write out the addition, multiplication, and "inverse" tables for each of the following fields? (a) \(\left[\mathbb{Z}_{2} ;+_{2}, \times_{2}\right]\) (b) \(\left[\mathbb{Z}_{3} ;+_{3}, \times_{3}\right]\) (c) \(\left[\mathbb{Z}_{5} ;+5, \times_{5}\right]\)

Review the definition of rings to show that the following are rings. The operations involved are the usual operations defined on the sets. Which of these rings are commutative? Which are rings with unity? For the rings with unity, determine the unity and all units. (a) \([Z ;+, \cdot]\) (b) \(\left[\mathrm{C}_{i}+_{1}+\right]\) (c) \(\left[\mathrm{Q} ;+,{ }^{*}\right]\) (d) \(\left[M_{2 \times 2}(\mathbb{R}) ;+,+\right]\) (e) \(\left[Z_{2} ;+_{2}, x_{2}\right]\)

\begin{aligned} &\text { Let } f(x)=\sum_{i=0}^{\infty} a_{i} x^{i} \text { and } g(x)=\sum_{i=0}^{\infty} b_{i} x^{i} \text { be elements of } R[[x]] . \text { Let }\\\ &f(x) \cdot g(x)=\sum_{i=0}^{\infty} d_{i} x^{i}=1 . \text { Apply basic algebra to }\left(a_{0}+a_{1} x+a_{2} x^{2}+\cdots\right)\\\ &\left(b_{0}+b_{1} x+b_{2} x^{2}+\cdots\right) \text { to derive the formula } d_{s}=\sum_{i=0}^{s} a_{i} b_{s-i} \text { for the co- }\\\ &\text { efficients of } f(x) \cdot g(x) \text { . Hence, to show that } f(x) \cdot g(x)=\sum_{s=0}^{\infty}\left(\sum_{i=0}^{s} a_{i} b_{s-i}\right) x^{s} \end{aligned}

Let \(f(x)=1+x\) and \(g(x)=1+x+x^{2}\). Compute the following sums and products in the indicated rings. (a) \(f(x)+g(x)\) and \(f(x) \cdot g(x)\) in \(\mathbb{Z}[x]\) (b) \(f(x)+g(x)\) and \(f(x) \cdot g(x)\) in \(\mathbb{Z}_{2}[x]\) (c) \((f(x) \cdot g(x)) \cdot f(x)\) in \(\mathbb{Q}[x]\) (d) \((f(x) \cdot g(x)) \cdot f(x)\) in \(\mathbb{Z}_{2}[x]\) (e) \(f(x) \cdot f(x)+f(x) \cdot g(x)\) in \(\mathbb{Z}_{2}[x]\)

(a) Determine the splitting field of \(f(x)=x^{2}+1\) over \(\mathbb{R}\). This means consider the polynomial \(f(x)=x^{2}+1 \in \mathbb{R}[x]\) and find the smallest field that contains \(\mathbb{R}\) and all the zeros of \(f(x) .\) Denote this field by \(\mathbb{R}(i)\) (b) \(\mathbb{R}(i)\) is more commonly referred to by a different name. What is it? (c) Show that \(\\{1, i\\}\) is a basis for the vector space \(\mathbb{R}(i)\) over \(\mathbb{R}\). What is the dimension of this vector space (over \(\mathbb{R}) ?\)

Determine the splitting field of \(x^{4}-5 x^{2}+6\) over \(\mathbb{Q}\).

Show that the following pairs of rings are not isomorphic: (a) \([\mathbb{Z} ;+, \cdot]\) and \(\left[M_{2 \times 2}(\mathbb{Z}) ;+_{+} \cdot\right]\) (b) \([3 Z ;+, \cdot \mid\) and \([4 Z ;+, \cdot]\)

Write out the operation tables for \(\mathbb{Z}_{2}^{2}\). Is \(\mathbb{Z}_{2}^{2}\) a ring? An integral domain? A field? Explain.

(a) Find all zeros of \(x^{4}+1\) in \(\mathbb{Z}_{3}\). (b) Find all zeros of \(x^{5}+1\) in \(\mathbb{Z}_{5}\).

(a) Show that \(x^{3}+x+1\) is irreducible over \(\mathbb{Z}_{2}\). (b) Determine the splitting field of \(x^{3}+x+1\) over \(\mathbb{Z}_{2}\). (c) By Theorem 16.2.10, you have described all fields of order \(2^{3}\).

Determine all values \(x\) from the given field that satisty the given equation: (a) \(x+1=-1\) in \(\mathbb{Z}_{2}, \mathbb{Z}_{3}\) and \(\mathbb{Z}_{5}\) (b) \(2 x+1=2\) in \(\mathbb{Z}_{3}\) and \(\mathbb{Z}_{5}\) (c) \(3 x+1=2\) in \(\mathbb{Z}_{5}\)

(a) Show that \(3 Z\) is a subring of the ring \([Z ;+,-]\) (b) Find all subrings of \(\mathbb{Z}_{8}\). (c) Find all subrings of \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\)

(a) Determine the inverse of \(h(x)=\sum_{i=0}^{\infty} 2^{i} x^{i}\) in \(\mathbb{Q}[[x]]\). (b) Use the procedures in Chapter 8 to find a rational generating function for \(h(x)\) in part a. Find the multiplicative inverse of this function.

Determine which of the following are reducible over \(\mathbb{Z}_{2}\). Explain. (a) \(f(x)=x^{3}+1\) (b) \(g(x)=x^{3}+x^{2}+x\). (c) \(h(x)=x^{3}+x^{2}+1\). (d) \(k(x)=x^{4}+x^{2}+1 .\) (Be careful.)

(a) Prove that if \(p\) and \(q\) are prime, then \(\mathbb{Z}_{p} \times \mathbb{Z}_{q}\) is never a field. (b) Can \(\mathbb{Z}_{p}{\underline{\phantom{xx}}}^{n}\) be a field for any prime \(p\) and any positive integer \(n \geq 2 ?\)

Let \(a(x)=1+3 x+9 x^{2}+27 x^{3}+\cdots=\sum_{i=0}^{\infty} 3^{i} x^{i}\) and \(b(x)=1+x+x^{2}+\). \(x^{3}+\cdots=\sum_{i=0}^{\infty} x^{i}\) both in \(\mathbb{R}[[x]]\). (a) What are the first four terms (counting the constant term as the \(0^{\text {th }}\) term \()\) of \(a(x)+b(x) ?\) (b) Find a closed form expression for \(a(x)\). (c) What are the first four terms of \(a(x) b(x) ?\)

Determine all solutions to the following equations over \(\mathbb{Z}_{2}\). That is, find all elements of \(\mathbb{Z}_{2}\) that satisfy the equations. (a) \(x^{2}+x=0\) (c) \(x^{3}+x^{2}+x+1=0\) (b) \(x^{2}+1=0\) (d) \(x^{3}+x+1=0\)

(a) Determine all solutions of the equation \(x^{2}-5 x+6=0\) in Z. Can there be any more than two solutions to this equation (or any quadratic equation) in \(Z ?\) (b) Find all solutions of the equation in part a in \(Z_{12}\). Why are there more than two solutions?

Write as an extended power series: (a) \(\left(x^{4}-x^{5}\right)^{-1}\) (b) \(\left(x^{2}-2 x^{3}+x^{4}\right)^{-1}\)

Solve the equation \(x^{2}+4 x+4=0\) in the following rings. Interpret 4 as \(1+1+1+1,\) where 1 is the unity of the ring. (a) in \(\mathbb{Z}_{8}\) (b) in \(M_{2 \times 2}(\mathbb{R})\) (c) in \(\mathbb{Z}\) (d) in \(\mathbb{Z}_{3}\)

Determine all zeros of \(x^{4}+3 x^{3}+2 x+4\) in \(\mathbb{Z}_{5}[x]\).

Let \(\mathbb{Q}(\sqrt{2})=\\{a+b \sqrt{2} \mid a, b \in \mathbb{Q}\\}\). (a) Prove that \([\mathbb{Q}(\sqrt{2}) ;+, \cdot]\) is a field. (b) Show that \(\mathbb{Q}\) is a subfield of \(\mathbb{Q}(\sqrt{2})\). For this reason, \(\mathbb{Q}(\sqrt{2})\) is called an extension field of \(\mathbb{Q}\). (c) Show that all the roots of the equation \(x^{2}-4 x+\frac{7}{2}=0\) lie in the extension field \(\mathbb{Q}(\sqrt{2})\). (d) Do the roots of the equation \(x^{2}-4 x+3=0\) lie in this field? Explain.

The relation "is isomorphic to \(^{\text {" }}\) on rings is an equivalence relation. Explain the meaning of this statement.

Show that \(x^{2}-3\) is irreducible over \(\mathbb{Q}\) but reducible over the field of real numbers.

(a) Prove that the ring \(\mathbb{Z}_{2} \times \mathbb{Z}_{3}\) is commutative and has unity. (b) Determine all zero divisors for the ring \(\mathbb{Z}_{2} \times \mathbb{Z}_{3}\). (c) Give another example illustrating the fact that the product of two integral domains may not be an integral domain. Is there an example where the product is an integral domain?

Boolean Rings. Let \(U\) be a nonempty set. (a) Verify that \([\mathcal{P}(U) ; \oplus, \cap]\) is a commutative ring with unity. (b) What are the units of this ring?

(a) Show that the field \(\mathbb{R}\) of real numbers is a vector space over \(\mathbb{R}\). Find a basis for this vector space. What is \(\operatorname{dim} \mathbb{R}\) over \(\mathbb{R} ?\) (b) Repeat part a for an arbitrary field F. (c) Show that \(\mathbb{R}\) is a vector space over \(\mathbb{Q}\).

(a) For any ring \([R ;+, \cdot]\), expand \((a+b)(c+d)\) for \(a, b, c, d \in R\). (b) If \(R\) is commutative, prove that \((a+b)^{2}=a^{2}+2 a b+b^{2}\) for all \(a, b \in R\)

(a) Let \(R\) be a commutative ring with unity. Prove by induction that for \(n \geq 1,(a+b)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l}n \\ k\end{array}\right) a^{k} b^{n-k}\) (b) Simplify \((a+b)^{5}\) in \(\mathbb{Z}_{5}\). (c) Simplify \((a+b)^{10}\) in \(\mathbb{Z}_{10}\).

Let \(U\) be a finite set. Prove that the Boolean ring \([\mathcal{P}(U) ; \oplus, \cap]\) is isomorphic to the ring \(\left[\mathrm{Z}_{2}^{n}:+,+\right] .\) where \(n=|U|\)