Chapter 24

\(A\left[x_{1}, x_{2}\right]\) denotes the ring of all the polynomials in two letters \(x_{1}\) and \(x_{2}\) with coefficients in \(A\). For example, \(x^{2}-2 x y+y^{2}+x-5\) is a quadratic polynomial in \(\mathbb{Q}[x, y] .\) More generally, \(A\left[x_{1}, \ldots, x_{n}\right]\) is the ring of all the polynomials in \(n\) letters \(x_{1}, \ldots, x_{n}\) with coefficients in \(A .\) Formally it is defined as follows: Let \(A\left[x_{1}\right]\) be denoted by \(A_{1} ;\) then \(A_{1}\left[x_{2}\right]\) is \(A\left[x_{1}, x_{2}\right]\). Continuing in this fashion, we may adjoin one new letter \(x_{i}\) at a time, to get \(A\left[x_{1}, \ldots, x_{n}\right]\) 1 Prove that if \(A\) is an integral domain, then \(A\left[x_{1}, \ldots, x_{n}\right]\) is an integral domain.

Let \(A\) and \(B\) be rings and let \(h: A \rightarrow B\) be a homomorphism with kernel \(K .\) Define \(\bar{h}: A[x] \rightarrow B[x]\) by: $$ \bar{h}\left(\dot{a}_{0}+a_{1} x+\cdots+a_{n} x^{n}\right)=h\left(a_{0}\right)+h\left(a_{1}\right) x+\cdots+h\left(a_{n}\right) x^{n} $$ (We say that \(\bar{h}\) is induced by \(h\).) 1 Prove that \(\bar{h}\) is a homomorphism from \(A[x]\) to \(B[x]\)

Homomorphisms of Domains of Polynomials Let \(A\) be an integral domain. 1 Let \(h: A[x] \rightarrow A\) map every polynomial to its constant coefficient; that is, $$ h\left(a_{0}+a_{1} x+\cdots+a_{n} x^{n}\right)=a_{0} $$ Prove that \(h\) is a homomorphism from \(A[x]\) onto \(A\), and describe its kernel.

Subrings and Ideals in \(A[x]\) 1 Show that if \(B\) is a subring of \(A\), then \(B[x]\) is a subring of \(A[x]\).

D. Domains \(A[x]\) where \(A\) Has Finite Characteristic In each of the following, let \(A\) be an integral domain. 1 Prove that if \(A\) has characteristic \(p\), then \(A[x]\) has characteristic \(p .\)

Rings \(A[x]\) where \(A\) Is Not an Integral Domain 1 If \(A\) is not an integral domain, neither is \(A[x]\). Prove this by showing that if \(A\) has divisors of zero, so does \(A[x]\)

Problems Involving Concepts and Definitions 1 Is \(x^{8}+1=x^{3}+1\) in \(\mathbb{Z}_{5}[x] ?\) Explain your answer.

Let \(a(x)=2 x^{2}+3 x+1\) and \(b(x)=x^{3}+5 x^{2}+x .\) Compute \(a(x)+b(x)\) \(a(x)-b(x)\) and \(a(x) b(x)\) in \(\mathbb{Z}[x], \mathbb{Z}_{5}[x], \mathbb{Z}_{6}[x]\), and \(\mathbb{Z}_{7}[x]\)

Give a reasonable definition of the degree of any polynomial \(p(x, y)\) in \(A[x, y]\) and then list all the polynomials of degree \(\leq 3\) in \(\mathbb{Z}_{3}[x, y]\). Let us denote an arbitrary polynomial \(p(x, y)\) in \(A[x, y]\) by \(\sum a_{i j} x^{i} y^{\prime}\) where \(\sum\) ranges over some pairs \(i, j\) of nonnegative integers.

If \(B\) is an ideal of \(A, B[x]\) is not necessarily an ideal of \(A[x] .\) Give an example to prove this contention.

Use part 1 to give an example of an infinite integral domain with finite characteristic.

Give examples of divisors of zero, of degrees 0,1, and 2 , in \(\mathbb{Z}_{4}[x]\).

Is there any ring \(A\) such that in \(A[x]\), some polynomial of degree 2 is equal to a polynomial of degree 4 ? Explain.

Find the quotient and remainder when \(x^{3}+x^{2}+x+1\) is divided by \(x^{2}+3 x+2\) in \(\mathbb{Z}[x]\) and in \(\mathbb{Z}_{5}[x] .\)

In \(\mathbb{Z}_{10}[x],(2 x+2)(2 x+2)=(2 x+2)\left(5 x^{3}+2 x+2\right)\), yet \((2 x+2)\) cannot be canceled in this equation. Explain why this is possible in \(\mathbb{Z}_{10}[x]\), but not in \(\mathbb{Z}_{5}[x]\).

Find the quotient and remainder when \(x^{3}+2\) is divided by \(2 x^{2}+3 x+4\) in \(\mathbb{Z}[x]\) in \(\mathbb{Z}_{3}[x]\), and in \(\mathbb{Z}_{5}[x]\) We call \(b(x)\) a factor of \(a(x)\) if \(a(x)=b(x) q(x)\) for some \(q(x)\), that is, if the remainder when \(a(x)\) is divided by \(b(x)\) is equal to zero.

Prove that deg \(a(x, y) b(x, y)=\operatorname{deg} a(x, y)+\operatorname{deg} b(x, y)\) if \(A\) is an integral domain.

Let \(g: A[x] \rightarrow A\) send every polynomial to the sum of its coefficients. Prove that \(g\) is a surjective homomorphism, and describe its kernel.

Let \(J\) consist of all the elements in \(A[x]\) whose constant coefficient is equal to zero. Prove that \(J\) is an ideal of \(A[x]\)

Prove that if \(A\) has characteristic \(p\), then in \(A[x],(x+c)^{p}=x^{p}+c^{p} .\) (You may use essentially the same argument as in the proof of the binomial theorem.)

Give examples in \(\mathbb{Z}_{4}[x]\), in \(\mathbb{Z}_{6}[x]\), and in \(\mathbb{Z}_{9}[x]\) of polynomials \(a(x)\) and \(b(x)\) such that \(\operatorname{deg} a(x) b(x)<\operatorname{deg} a(x)+\operatorname{deg} b(x)\)

Let \(A\) be an integral domain; prove the following: If \((x+1)^{2}=x^{2}+1\) in \(A[x]\), then \(A\) must have characteristic \(2 .\) If \((x+1)^{4}=x^{4}+1\) in \(A[x]\), then \(A\) must have characteristic \(2 .\) If \((x+1)^{6}=x^{6}+2 x^{3}+1\) in \(A[x]\), then \(A\) must have characteristic 3

Show that the following is true in \(A[x]\) for any ring \(A:\) For any odd \(n\), (a) \(x+1\) is a factor of \(x^{n}+1\) (b) \(x+1\) is a factor of \(x^{n}+x^{n-1}+\cdots+x+1\)

If \(A\) is an integral domain, we have seen that in \(A[x]\), $$ \operatorname{deg} a(x) b(x)=\operatorname{deg} a(x)+\operatorname{deg} b(x) $$ Show that if \(A\) is not an integral domain, we can always find polynomials \(a(x)\) and \(b(x)\) such that \(\operatorname{deg} a(x) b(x)<\operatorname{deg} a(x)+\operatorname{deg} b(x)\)

Find an example of each of the following in \(\mathbb{Z}_{8}[x]:\) a divisor of zero, an invertible element, an idempotent element.

Prove the following: In \(\mathbb{Z}_{3}[x], x+2\) is a factor of \(x^{m}+2\), for all \(m .\) In \(\mathbb{Z}_{n}[x]\), \(x+(n-1)\) is a factor of \(x^{m}+(n-1)\), for all \(m\) and \(n\).

If \(h: \mathbb{Z} \rightarrow \mathbb{Z}_{n}\) is the natural homomorphism, let \(\bar{h}: \mathbb{Z}[x] \rightarrow \mathbb{Z}_{n}[x]\) be the homomorphism induced by \(h\). Prove that \(\bar{h}(a(x))=0\) iff \(n\) divides every coefficient of \(a(x)\).

Show that if \(A\) is an integral domain, the only invertible elements in \(A[x]\) are the constant polynomials \(\pm 1\). Then show that in \(\mathbb{Z}_{4}[x]\) there are invertible polynomials of all degrees.

Explain why \(x\) cannot be invertible in any \(A[x]\), hence no domain of polynomials can ever be a field.

Prove that there is no integer \(m\) such that \(3 x^{2}+4 x+m\) is a factor of \(6 x^{4}+50\) in \(\mathbb{Z}[x]\)

Give all the ways of factoring \(x^{2}\) in \(\mathbb{Z}_{9}[x]\); in \(\mathbb{Z}_{5}[x] .\) Explain the difference in behavior.

There are rings such as \(P_{3}\) in which every element \(\neq 0,1\) is a divisor of zero. Explain why this cannot happen in any ring of polynomials \(A[x]\), even when \(A\) is not an integral domain.

Find all the square roots of \(x^{2}+x+4\) in \(\mathbb{Z}_{5}[x]\). Show that in \(\mathbb{Z}_{8}[x]\). there are infinitely many square roots of \(1 .\)

Show that in every \(A[x]\), there are elements \(\neq 0,1\) which are not idempotent, and elements \(\neq 0,1\) which are not nilpotent.