189-456A: Algebra 3

Assignment 4

Due: Wednesday, November 6.

The first three exercises aim to prove Fermat's ``Christmas theorem", that every prime of the form $1+4k$ can be written as the sum of two squares of integers.

1. Let $R$ be the ring ${\mathbb Z}[i]$ of Gaussian integers, consisting of elements of the form $x+yi$ with $i^2=-1$ and $x,y\in{\mathbb Z}$. Define the norm of $x+yi$ to be $N(x+yi)= x^2+y^2$.
Show that $R$ satisfies the euclidean division algorithm with remainder, i.e., that for all $a,b\in R$, there are $q,r\in R$ with $a=bq+r$ and $N(r)\lt N(b)$. Use this to show that every ideal in $R$ is principal.

2. Let $p$ be a prime number which is of the form $1+4k$. Show that $-1$ is the square of an element in the ring ${\mathbb Z}/p{\mathbb Z}$. (Hint: think about the structure of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^\times$, and its elements of order $2$ and $4$.)

Letting $t$ be an integer such that $p$ divides $t^2+1$, show that the ideal of $R$ generated by $p$ and $t+i$ is a proper ideal in $R$, i.e, not equal to $R$.

3. Show that every prime $p$ pf the form $1+4k$ can be expressed as a sum of two squares. (Hint: consider a generator of the non-trivial ideal $(p, t+i)$ considered in question 2.)

The goal of the next six questions is to adapt the above proof to the setting of a non-comutative ring, the ring of Hurwitz integer quaternions, in order to prove the celebrated ``four squares" theorem of Lagrange, that every positive integer can be expressed as a sum of four integer squares.

4. 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 ${\rm norm}(z)$.

5. 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.

6. 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 ${\rm norm}(r) \lt {\rm norm}(b)$.

7. 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 Q6 to prove that any left ideal $I$ in $R$ is principal, i.e., that it is of the form $R b$ for a suitable $b\in R$.

8. 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.)

9. Using the results you have obtained in Q7 and Q8, 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.

10. 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 $T$.