189-456A: Algebra 3

Assignment 4

Due: Wednesday, November 8.

In class, we proved a celebrated theorem of Fermat asserting that any prime of the form $1+4k$ is a sum of two perfect squares, based on the fact that the ring of Gaussian integers is a principal ideal domain. Questions 1 to 6 aim to give a proof, in a similar spirit, of a theorem of Lagrange which asserts that all positive integers can be written as a sum of four perfect squares.

1. Recall the ring $${\mathbb H}:= \{ a + b i + cj + dk \mbox{ with } a,b,c,d\in \mathbb R \} $$ of Hamilton quaternions, whose multiplication table is determined by the rules $$ i^2 = j^2 = k^2 =-1, \quad ij =-ji = k, \quad jk = -kj = i, \quad ki =-ik = j.$$ The conjugate of a quaternion $z=a+bi+cj+dk$ is the quaternion $\bar z := a-bi-cj-dk$. Show that $$ \overline{z_1 z_2} = \bar z_2 \bar z_1, \quad z \bar z = \bar z z = a^2+b^2+c^2+d^2,$$ and use this to conclude that ${\mathbb H}$ is a skew field, i.e., a non-commutative ring in which every non-zero element $z$ has a multiplicative inverse $z'$ satisfying $z'z = zz' = 1$.

The positive real number $z\bar z$ is called the norm of $z$, and is denoted ${\rm norm}(z)$.

2. Let $R$ be the subset of ${\mathbb H}$ defined by $$ R := \left\{ a+bi+cj+dk \mbox{ with } a,b,c,d\in \mathbb Z \mbox{ or } a,b,c,d\in \frac{1}{2}+\mathbb Z. \right\}$$ Show that $R$ is a subring of ${\mathbb H}$, and that the norm of any element of $R$ is an integer. The ring $R$, which was first seriously studied by the German mathematician Adolf Hurwitz in 1919, is commonly called the ring of Hurwitz integer quaternions.

3. Show that $R$ admits a right Euclidean division algorithm, i.e., that for all $a,b\in R$, there exists $q, r\in R$ for which $a = qb + r$ with ${\rm norm}(r) < {\rm norm}(b)$.

4. A left ideal in $R$ is an additive subgroup of $R$ which is closed under left multiplication by elements of $R$. Use the result you showed in part 3 to prove that any left ideal $I$ in $R$ is principal, i.e., that it is of the form $R b$ for a suitable $b\in R$.

5. Let $p$ be a prime number. Show that $R$ contains an element $z$ whose norm is divisible by $p$ but not by $p^2$. (Hint: apply the pigeon-hole principle to the two subsets $$ A = \{ 1+a^2 \mbox{ with } a \in \mathbb Z/p\mathbb Z\}, \qquad B = \{ -b^2 \mbox{ with } b \in \mathbb Z/p\mathbb Z\} $$ to obtain a quaternion of the form $1+ai +bj$ with the desired properties.)

6. Using the results you have obtained in Q4, show that every prime number can be written as the sum of four integer squares. Conclude that the same is true of any positive integer, by using the multiplicativity of the norm (i.e.,, that ${\rm norm}(z_1z_2) = {\rm norm}(z_1) {\rm norm}(z_2)$).

The goal of this question is to revisit the statement that the centraliser of an element of order 7 in $GL_3(F_2)$ has order 7, using some ideas of ring theory and linear algebra.

7. Let $G= GL_3(F_2)$ be the group of order $168$ that has occupied us already in earlier assignments, acting on the three-dimensional space $V=F_2^3$ of column vectors with entries in the field $F_2$ with two elements.

Let $T$ be an element of $G$ of order $7$. Show that the subring $F\subset M_3(F_2)$ of the ring of endomorphisms of $V$ generated by $T$ is a field with $8$ elements. (Hint: show that $T$ satisfies an irreducible polynomial $p(x)$ of degree $3$ over $F_2$.) Show that the action of $T$ endows $V$ with the structure of a one-dimensional vector space over $F$, and that the linear transformations of $V$ that commute with $T$ are precisely those that are $F$-linear. Use this to conclude that the centralizer of $T$ in $G$ is the group of order $7$ generated by $T$.

8. Determine which of the following ideals in $\mathbb Z[i]$ are prime, by describing the associated quotient: $$I_1=(2), \qquad I_2=(1+i), \qquad I_3 = (5), \qquad I_4 = (7), \qquad I_5 = (4+i), \qquad I_6=(5+i).$$

Hensel's Lemma.

9. Let $f(x)\in \mathbb Z[x]$ be a polynomial with integer coefficients and let $a_1$ be an integer for which $$ f(a_1) = 0 \pmod{p}, \qquad f'(a_1) \ne 0 \pmod{p}.$$ Show that there are elements $a_n\in \mathbb Z$ for each $n\ge 2$ satisfying $$ f(a_n) \equiv 0 \pmod{p^n}, \qquad a_n \equiv a_{n-1} \pmod{p^{n-1}}.$$ (Hint: recall from your analysis or numerical analysis course the method of Newton iteration for finding the zeroes of a polynomial by successive approximation.)

10. Let $R$ be a commutative ring that is either finite as a set, or that contains a field over which it is a finite-dimensional vector space. Show that $R$ is an integral domain if and only if it is a field. (Hint: given $a\in R-\{0\}$, consider the map from $R$ to $R$ sending $x$ to $ax$.)