Chapter 21

Prove that if \(p\) is prime and \(0

For the prime \(p=7\), calculate for each integer \(a\) with \(1<\) \(a<7\) the smallest exponent \(m\) such that \(a^{m} \equiv 1(\bmod 7)\). Show that the theorem in Exercise 1 holds for all \(a\).

Determine the primitive roots of \(p=13\), that is, determine numbers \(a\) for which \(p-1\) is the smallest exponent such that \(a^{p-1} \equiv 1(\bmod p)\).

Complete Gauss's determination that 453 is a quadratic residue modulo 1236 by showing that a. If \(x^{2} \equiv 453(\bmod 4), x^{2} \equiv 453(\bmod 3)\), and \(x^{2} \equiv 453\) (mod 103) are all solvable, then so is \(x^{2} \equiv 453\) (mod 4 . \((3 \cdot 103)\). b. 453 is a quadratic residue modulo both 4 and 3 . c. \(\left(\frac{453}{103}\right)=\left(\frac{41}{103}\right)\). d. \(\left(\frac{5}{41}\right)=1\).

Show that the Gaussian integer \(a+b i(b \neq 0)\) is prime if and only if the \(\operatorname{norm} a^{2}+b^{2}\) is an ordinary prime.

Show that if any Gaussian prime \(p\) divides the product \(a b c \cdots\) of Gaussian primes, then \(p\) must equal one of those primes, or one of them multiplied by a unit. (Hint: Take norms of both sides.)

Use Germain's theorem to show that if there is a solution to the Fermat equation for exponent 3, then one of \(x, y\), or \(z\) must be divisible by 9 . To do this show, first, that 3 is not a cube modulo 7 and, second, that no two nonzero third-power residues modulo 7 differ by 1 .

Show that in the domain of integers of the form \(a+\) \(b \sqrt{-17}\), Liouville's factorization \(169=13 \cdot 13=(4+\) \(3 \sqrt{-17})(4-3 \sqrt{-17})\) in fact demonstrates that unique factorization into primes fails in that domain. (Hint: Use norms to show that each of the four factors is irreducible.)

Show that the Gaussian integers form a Euclidean domain. That is, show that, given two Gaussian integers \(z, m\), there exist two others, \(q, r\), such that \(z=q m+r\) and \(N(r)<\) \(N(m)\)

Show that in the domain of complex integers of the form \(a+b \sqrt{-5}\), the integers \(2,3,-2+\sqrt{-5},-2-\sqrt{-5}, 1+\) \(\sqrt{-5}, 1-\sqrt{-5}\) are all irreducible.

Determine the cosets of the cyclic subgroup of order 6 of the cyclic group of order 18 .

Use Gauss's method to solve the cyclotomic equation \(x^{6}+\) \(x^{5}+x^{4}+x^{3}+x^{2}+x+1=0\)

In the example dealing with Gauss's solution to \(x^{19}-1=0\), show that \(\beta_{1}, \beta_{8}\), and \(\beta_{7}\) are roots of the cubic equation \(x^{3}-\alpha_{1} x^{2}+\left(\alpha_{2}+\alpha_{4}\right) x-2-\alpha_{2}=0\), where the \(\alpha\) 's and \(\beta\) 's are as in the text.

In the example dealing with Gauss's solution to \(x^{19}-1=0\), show that \(r\) and \(r^{18}\) are both roots of \(x^{2}-\beta_{1} x+1=0\), where \(r\) and \(\beta_{1}\) are as in the text.

Calculate the Galois group \(G\) of the equation \(x^{3}+6 x=\) 6 over the rational numbers. Show that this group has a normal subgroup \(H\) such that both \(H\) and the index of \(H\) in \(G\) are primes.

Show that the Galois group of the equation \(x^{5}-2\) over the rational numbers can be expressed as the group of substitutions of the form \(x^{\prime} \equiv a x+b(\bmod 5)\) and therefore has 20 elements.

Find a fifth-degree polynomial that is not solvable by radicals.

Show that the order of the group \(S L(2, p)\) is \(p\left(p^{2}-1\right)\) and that of \(P S L(2, p)\) is \(\frac{1}{2} p\left(p^{2}-1\right)\).

Show that Hamilton's laws of operation on number couples \((\alpha, \beta)\) mirror the analogous laws of operation on complex numbers \(\alpha+\beta i\).

Let \(\alpha=3+4 i+7 j+k\) and \(\beta=2-3 i+j-k\) be quaternions. Calculate \(\alpha \beta\) and \(\alpha / \beta\).

Define the modulus \(|\alpha|\) of a quaternion \(a+b i+c j+d k\) by \(|\alpha|=a^{2}+b^{2}+c^{2}+d^{2}\). Show that \(|\alpha \beta|=|\alpha||\beta|\).

Interpret the remaining three of Boole's equations \(x \bar{y} z=0\), \(x \bar{y} \bar{z}=0, \bar{x} y z=0\) from the case in the text, where \(x\) stands for clean beasts, \(y\) for beasts that divide the hoof, and \(z\) for beasts that chew the cud.

Show that if the substitution \(x=\alpha x^{\prime}+\beta y^{\prime}, y=\gamma x^{\prime}+\) \(\delta y^{\prime}\) with \(\alpha \delta-\beta \gamma=1\) transforms the quadratic form \(F=\) \(a x^{2}+2 b x y+c y^{2}\) into the form \(F^{\prime}=a^{\prime} x^{\prime 2}+2 b^{\prime} x^{\prime} y^{\prime}+\) \(c^{\prime} y^{\prime 2}\), then there is an "inverse" substitution of the same form that transforms \(F^{\prime}\) into \(F\).

Prove that if the product of two matrices is the zero matrix, then at least one of the factors has determinant 0 .

Show explicitly the truth of the Cayley-Hamilton theorem that a matrix \(A\) satisfies its characteristic equation \(\operatorname{det}(A-\) \(\lambda I)=0\) in the case where \(A\) is a \(2 \times 2\) matrix.

Show that the matrix $$ L=\left(\begin{array}{cc} \frac{a+Y}{X} & \frac{b}{X} \\ \frac{c}{X} & \frac{d+Y}{X} \end{array}\right) $$ where \(X=\sqrt{a+d+2 \sqrt{a d-b c}}\) and \(Y=\sqrt{a d-b c}\), is the square root of the matrix $$ M=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) $$

Use Cauchy's technique to find an orthogonal substitution that converts the quadratic form \(2 x^{2}+6 x y+5 y^{2}\) into a sum or difference of squares.

Solve explicitly the system of linear equations $$ \begin{aligned} &2 u+v+2 x+y+3 z=0 \\ &5 u+3 v-4 x+3 y-6 z=0 \\ &u+v-8 x+y-12 z=0 \end{aligned} $$ First determine the order of the maximal nonvanishing determinant in the matrix of coefficients.

Show that two equivalent quadratic forms have the same discriminant.

Given a group of order \(p q,(p>q)\), with \(S^{q}=1\) and \(T^{p}=\) 1, show that if \(S^{-1} T S=T^{r}\), then \(r^{q} \equiv 1(\bmod p)\).

Create a field of order \(5^{3}\) by finding a third-degree irreducible congruence modulo 5 .

Compare Weber's definition of a field with the standard modern definition. Can some of Weber's axioms be proved from other ones?

Try to create a multiplication for number triples, written, say, in the form \(\alpha+\beta i+\gamma j\), which satisfies Hamilton's criteria for a reasonable multiplication. Namely, the multiplication must satisfy the commutative and associative laws, must be distributive over addition, must allow unique division, and must satisfy the modulus multiplication rule. What problems do you run into?

Design a lesson explaining negative numbers using either Peacock's principle of permanence of equivalent forms or Hamilton's formulation via pairs of positive numbers. Which formulation would work better in a classroom? Why?