Cyclotomics, The Fundamental Unit in ζ8

The Fundamental Unit in ζ8

Let y be the 8th root of 1, generating a 4 dimensional extension of the rationals (or the integers). The cyclotomic polynomial is x4+1 = 0. The basis of the extension is 1, y, y2, and y3. Perform polynomial arithmetic as usual, but replace y4 with -1 wherever it appears.

The dirichlet unit theorem tells us there is one fundamental unit, along with the 8 roots of 1. Let's see if we can find it.

Let u = 1+y+y2. This is s times 1+i, where s is 1+sqrt(½). Therefore |u| = 1+sqrt(2). Clearly u lies outside the unit circle.

Let v = 1-y2+y3, and show that uv = 1. Thus u and v are units. Powers of u spiral outward around the complex plane towards infinity, while powers of v spiral in towards the origin.

A ring automorphism implements an automorphism on the group of units. It carries the fundamental unit to another fundamental unit. In this case the unit is multiplied by a power of y, and it may be inverted. These are the only possibilities, since there is but one fundamental unit. We'll use this fact below.

If u is not fundamental then u = gn for some generator g and some integer n > 1. This means |g| is the nth root of |u|, and |g| ≤ sqrt(|u|). We know that |u| = 2.414, so |g| is smaller than 1.554. We will show that |g| cannot be this small, for g a unit outside the unit circle. That will prove u is fundamental.

Given any element a+by+cy2+dy3, let r be the maximum absolute value of the coefficients. Divide by an appropriate power of y, so that r is the real coefficient. Negate if necessary, so that a is positive. Now a = r.

If re(by+dy3) < 0, consider the conjugate produced by y → y5. Now we know that by+dy3 contributes positively to the real component. It may work together with cy2 to introduce a significant imaginary component, but we can't count on that. Therefore this conjugate of our original element is at least distance r from the origin.

Let g be the fundamental unit outside the unit circle, and let r be its largest coefficient. Multiply by a power of y, so that a = r. Applying an automorphism, as described above, keeps |g| the same, or inverts it, since g is the fundamental unit. It also increases |g|, which is a contradiction. Hence there is no need to apply such an automorphism. We already have re(by+dy3) ≥ 0, and the previous lemma tells us |g| ≥ r.

Since |g| is bounded by 1.554, none of its coefficients can have an absolute value beyond 1. Everything is ±1 or 0.

Of course g is nonzero. Rotate g, and multiply by -1 if necessary, so that a = 1.

If b = d = 0 we have 1±y2, an element that is 1.414 from the origin. This is below our bound of 1.554, but there is no integer n such that 1.414n = 2.414, hence g cannot be an nth root of u. Either b or d is nonzero.

If b = 0, select an isomorphism that maps y3 onto y, or y5, so that b = 1. If b was -1, replace y with -y, so that b becomes 1. In either case, a = b = 1.

If d = -1 g is 2.414 distance from the origin, not counting the contribution of cy2. That's too big.

If d = 1, and c = 0, |g| = sqrt(3), which is too large. Set c = -1 to minimize distance. This gives |g| = 1.082, but again, there is no n such that |g|n = |u|. Therefore d does not equal 1. Combine with the previous paragraph, and d = 0.

When c = 1 we have g = u. When c = 0, |g| = 1.847, and when c = -1, |g| = 1.732. That's all the cases, hence u is fundamental.

Every unit in the extension is some power of u (possibly negative) times some power of y (from 0 to 7).

Note that u has a real associate, namely u/y = sqrt(2)+1.