Minkowski’s linear forms theorem can be used to give sharper estimates for a positive real number *m* for which every ideal I contains an element *a* such that |N(*a*)| <= *m**N(I). The estimate I used in my first proof is very big, and unwieldy for concrete computations of class groups.

First, a very easy and still pretty bad *m* is SQRT(|d(K)|). The ideal I has some integral basis *x*1, *x*2, …, *x*(*n*), whose matrix by definition has determinant *d* = N(I)*SQRT(|d(K)|). We can let *t*1, *t*2, …, *t*(*n*) be any numbers we like that satisfy the conditions of the linear forms theorem, say *t*(*i*) = *d*^(1/*n*) for all *i*. There exists integers *a*(*j*) not all zero such that *w*(*i*)(*a*1*x*1 + … + *a*(*n*)*x*(*n*)) <= *t*(*i*) for all *i*, so that the element they define has norm at most *t*1**t*2*…**t*(*n*) = N(I)*SQRT(d(K)).

Recall from the proof of the finiteness of the class group that this shows that every ideal class of K contains a representative with norm <= SQRT(|d(K)|).

For a more sophisticated bound, first write *n* as *r* + 2*s*, where *r* is the number of real conjugates of K and *s* is the number of pairs of proper complex conjugates (proper complex ring homomorphisms always come in complex conjugate pairs). Then label the ring homomorphisms *w*(*i*) in such a way that the first *r* homomorphisms are real, and the next 2*s* are not, with *w*(*r*+1) = __ w(r+s+1)__,

*w*(

*r*+2) =

__, …,__

*w*(*r*+*s*+2)*w*(

*r*+

*s*) =

__.__

*w*(*r*+2*s*)Now, let *f* be a linear transformation from K = **Q**^*n* to **R**^*n* that sends *a* to (*w*1(*a*), *w*2(*a*), …, *w*(*r*)(*a*), Re(*w*(*r*+1)(*a*)), …, Re(*w*(*r*+*s*)(*a*)), Im(*w*(*r*+1)(*a*)), …, Im(*w*(*r*+*s*)(*a*))). If a complex number is equal to *a* + *bi* where *a* and *b* are real, then Re(*a* + *bi*) = *a* and Im(*a* + *bi*) = *b*. Note that the matrix corresponding to this transformation will have determinant with absolute value SQRT(|d(K)|)/2^*s* since the row operations that take *v* and * v* to Re(

*v*) and Im(

*v*) halve the determinant.

In **R**^*n*, let the region D consist of all points that in a way correspond to numbers in K with norm less than 1, i.e. points (*x*1, *x*2, …, *x*(*n*)) such that |*x*1|*|*x*2|*…*|*x*(*r*)|*|*x*(*r*+1)^2 + *x*(*r*+*s*+1)^2|*…*|*x*(*r*+*s*)^2 + *x*(*r*+2*s*)^2| <= 1. D is only convex if *r* = 1 and *s* = 0 or *r* = 0 and *s* = 1. Otherwise, we need to take some subregions of D.

If S is a convex region contained in D, and the volume of S is *k*, then we can get a bound of (2^(*r*+*s*)/*k*)*SQRT(|d(K)|). To see why, first let *y*1, *y*2, …, *y*(*n*) be an integral basis of the ideal I. The matrix corresponding to the basis has determinant of absolute value SQRT(|d(K)|)*N(I); the matrix whose (*i*, *j*)th entry is *x*(*i*)(*y*(*j*)), i.e. the transformation defined by *x*(*i*) applied to *y*(*j*), then has determinant of absolute value SQRT(|d(K)|)*N(I)/2^*s*.

Since the determinant is nonzero, the vectors *f*(*y*1), *f*(*y*2), …, *f*(*y*(*n*)) are linearly independent over **R**, so they define a lattice of volume SQRT(|d(K)|)*N(I)/2^*s*. Let *u* be the *n*th root of (2^(*r*+*s*)/*k*)*SQRT(|d(K)|)*N(J). Then the region *u*S has volume (*u*^*n*)*k* = (2^(*r*+*s*))*SQRT(|d(K)|)*N(J) = (2^*n*)*SQRT(|d(K)|)*N(I)/2^*s*.

This means that for every *u*‘ > *u*, there exists some nonzero point of the lattice defined by the *f*(*y*(*i*))’s, call it *v*, in *u*‘S. Since for a fixed *u*‘ there are finitely many such points, we can even find a point of the lattice in *u*S. That means *u*S contains a point of the form *f*(*a*) where *a* is in I. All points in S have norm at most 1, so |N(*a*)| <= *u*^*n* = (2^(*r*+*s*)/*k*)*SQRT(|d(K)|)*N(J).

Here I’m going to do a little bit of handwaving. A suitable region S that gives a good bound is the set of all points in **R**^*n* such that |*x*1| + |*x*2| + … + |*x*(*r*)| + 2*SQRT(*x*(*r*+1)^2 + *x*(*r*+*s*+1)^2) + … + 2*SQRT(*x*(*r*+*s*)^2 + *x*(*r*+2*s*)^2) <= *n*. S is fairly self-evidently convex and symmetric, and is contained in D because a theorem that says the geometric average of positive real numbers is at most their arithmetic average (i.e. SQRT(*ab*) <= (*a*+*b*)/2).

It’s possible to prove by induction on *r* and *s* that the volume of S is (2^*r*)*((pi/2)^*s*)*(*n*^*n*)/*n*!, so that we have a bound *m* = (4/pi)^*s***n*!/*n*^*n*. This complicated monster is called the Minkowski bound. Although deriving it is hard, it makes proving unique factorization in rings of integers a lot easier.

For example, take **Q**(SQRT(14)). The original *m* with respect to the integral basis {1, SQRT(14)} would give (1 + SQRT(14)^2 = 15 + 2*SQRT(14) = 22.48. With respect to the somewhat cleverer choice {1, 4-SQRT(14)}, we get exactly 11. The Minkowski bound gives SQRT(d(K))/2 = SQRT(14) = 3.74. As 2 = (4+SQRT(14))(4-SQRT(14)), the ideal 2 splits into two prime ideals. The ideal (3) is prime since if 3 divides (*a *+ *b**SQRT(14))(*c* +* d**SQRT(14)) without dividing any of *a*, *b*, *c*, or *d*, we get that *ac* + 14*bd* and *ad* + *bc* are divisible by 3. Modulo 3, 1 and 2 both have the square 1, so *ac* = *bd* mod 3 implies *abcd* = 1 mod 3, and *ad* + *bc* = 0 mod 3 implies *abcd* = -1 mod 3, a contradiction.