Chapter 26

Find three polynomials in \(\mathbb{Z}_{5}[x]\) which determine the same function as $$ x^{2}-x+1 $$

Prove that for \(k=1, \ldots, n:\) $$ a_{k}\left(x^{k}-c^{k}\right)=a_{k}(x-c)\left(x^{k-1}+x^{k-2} c+\cdots+c^{k-1}\right) $$

Let \(F\) be any field. Explain why, if \(a(x)\) is a quadratic or cubic polynomial in \(F[x]\), \(a(x)\) is irreducible in \(F[x]\) iff \(a(x)\) has no roots in \(F\).

Show that each of the following polynomials is irreducible over \(\mathbb{Q}\) : $$ \begin{array}{r} 3 x^{4}-8 x^{3}+6 x^{2}-4 x+6 ; \quad \frac{2}{3} x^{5}+\frac{1}{2} x^{4}-2 x^{2}+\frac{1}{2} ; \\ \frac{1}{5} x^{4}-\frac{1}{3} x^{3}-\frac{2}{3} x+1 ; \quad \frac{1}{2} x^{4}+\frac{4}{3} x^{3}-\frac{2}{3} x^{2}+1 \end{array} $$

The remainder of \(p(x)\), when divided by \(x-c\), is \(p(c)\).

Find all the rational roots of the following polynomials, and factor them into irreducible polynomials in \(Q[x]:\) $$ \begin{array}{r} 9 x^{3}+18 x^{2}-4 x-8 ; \quad 4 x^{3}-3 x^{2}-8 x+6 \\ 2 x^{4}+3 x^{3}-8 x-12 ; \quad 6 x^{4}-7 x^{3}+8 x^{2}-7 x+2 \end{array} $$

Find all the roots of the following polynomials in \(\mathbb{Z}_{5}[x]\), and factor the polynomials: $$ x^{3}+x^{2}+x+1 ; \quad 3 x^{4}+x^{2}+1 ; \quad x^{5}+1 ; \quad x^{4}+1 ; \quad x^{4}+4 $$

Prove that \(x^{4}+10 x^{3}+7\) is irreducible in \(\mathbb{Q}[x]\) by using the natural homomorphism from \(\mathbb{Z}\) to \(\mathbb{Z}_{5}\).

Prove that the following polynomials are irreducible in \(\mathbb{Q}[x]:\) $$ \frac{1}{2} x^{3}+2 x-\frac{3}{2} ; \quad 3 x^{2}-2 x-4 ; \quad x^{3}+x^{2}+\frac{3}{2} x+\frac{1}{2} ; \quad x^{3}+\frac{1}{2} ; \quad x^{2}-\frac{5}{2} x+\frac{3}{2} $$

It often happens that a polynomial \(a(y)\), as it stands, does not satisfy the conditions of Eisenstein's criterion, but with a simple change of variable \(y=x+c\), it does. It is important to note that if \(a(x)\) can be factored into \(p(x) q(x)\), then certainly \(a(x+c)\) can be factored into \(p(x+c) q(x+c)\). Thus, the irreducibility of \(a(x+c)\) implies the irreducibility of \(a(x)\) (a) Use the change of variable \(y=x+1\) to show that \(x^{4}+4 x+1\) is irreducible in \(\mathbb{Q}[x]\). [In other words, test \((x+1)^{4}+4(x+1)+1\) by Eisenstein's criterion.] (b) Find an appropriate change of variable to prove that the following are irreducible in \(\mathbb{[ x ]}\) $$ x^{4}+2 x^{2}-1 ; \quad x^{3}-3 x+1 ; \quad x^{4}+1 ; \quad x^{4}-10 x^{2}+1 $$

Let \(F\) be a field. Prove that each of the following is true in \(F[x] .\) \((x-c) \mid(p(x)-p(c))\)

Use Fermat's theorem to find all the roots of the following polynomials in \(\mathbb{Z}_{7}[x]:\) $$ x^{100}-1 ; \quad 3 x^{98}+x^{19}+3 ; \quad 2 x^{74}-x^{55}+2 x+6 $$

Prove that there is one and only one polynomial \(p(x)\) of degree \(\leq n\) such that \(p\left(a_{0}\right)=b_{0}, \ldots, p\left(a_{n}\right)=b_{n}\)

Suppose a monic polynomial \(a(x)\) of degree 4 in \(F[x]\) has no roots in \(F\). Then \(a(x)\) is reducible iff it is a product of two quadratics \(x^{2}+a x+b\) and \(x^{2}+c x+d\), that is, iff $$ a(x)=x^{4}+(a+c) x^{3}+(a c+b+d) x^{2}+(b c+a d) x+b d $$ If the coefficients of \(a(x)\) cannot be so expressed (in terms of any \(a, b, c, d \in F)\) then \(a(x)\) must be irreducible.

Prove that for any prime \(p, x^{p-1}+x^{p-2}+\cdots+x+1\) is irreducible in \(\mathbb{Q}[x]\) [HINT: By elementary algebra, hence $$ \begin{gathered} (x-1)\left(x^{p-1}+x^{p-2}+\cdots+x+1\right)=x^{p}-1 \\ x^{p-1}+x^{p-2}+\cdots+x+1=\frac{x^{p}-1}{x-1} \end{gathered} $$ Use the change of variable \(y=x+1\), and expand by the binomial theorem.

Using Fermat's theorem, find polynomials of degree \(\leq 6\) which determine the same functions as the following polynomials in \(\mathbb{Z}_{7}[x]\) : $$ 3 x^{75}-5 x^{54}+2 x^{13}-x^{2} ; 4 x^{108}+6 x^{101}-2 x^{81} ; 3 x^{103}-x^{73}+3 x^{55}-x^{25} $$

Use the Lagrange interpolation formula to prove that if \(F\) is a finite field, every function from \(F\) to \(F\) is equal to a polynomial function. (In fact, the degree of this polynomial is less than the number of elements in \(F\).)

Let \(a(x)\) and \(b(x)\) be in \(F[x]\). If \(a(x)\) and \(b(x)\) determine the same function, and if the number of elements in \(F\) exceeds the degree of \(a(x)\) as well as the degree of \(b(x)\), then \(a(x)=b(x)\)

Prove that the following polynomials are irreducible in \(\mathbb{Z}_{5}[x]:\) $$ 2 x^{3}+x^{2}+4 x+1 ; \quad x^{4}+2 ; \quad x^{4}+4 x^{2}+2 ; \quad x^{4}+1 $$

If \(a(x)\) and \(b(x)\) have the same roots in \(F\), are they necessarily associates? Explain.

If \(t(x)\) is any polynomial in \(F[x]\), and \(a_{0}, \ldots, a_{n} \in F\), the unique polynomial \(p(x)\) of degree \(\leq n\) such that \(p\left(a_{0}\right)=t\left(a_{0}\right), \ldots, p\left(a_{n}\right)=t\left(a_{n}\right)\) is called the Lagrange interpolator for \(t(x)\) and \(a_{0}, \ldots, a_{n} .\) Prove that the remainder, when \(t(x)\) is divided by \(\left(x-a_{0}\right)\left(x-a_{1}\right) \cdots\left(x-a_{n}\right)\), is the Lagrange interpolator.

Does \(2 x^{4}+3 x^{2}-2\) have any rational roots? Can it be factored into two polynomials of lower degree in \(\mathbb{Q}[x]\) ? Explain.

If \(a(x)\) is a monic polynomial of degree \(n\), and \(a(x)\) has \(n\) roots \(c_{1}, \ldots, c_{n} \in F\), then \(a(x)=\left(x-c_{1}\right) \cdots\left(x-c_{n}\right)\)

Suppose \(a(x)\) and \(b(x)\) have degree \(

There are infinitely many irreducible polynomials in \(\mathbb{Z}_{5}[x]\).

How many roots does \(x^{2}-x\) have in \(\mathbb{Z}_{10} ?\) In \(\mathbb{Z}_{11}\) ? Explain the difference.