Since childhood you have trusted a single fact without question: every whole number breaks into primes in exactly one way. That is the fundamental theorem of arithmetic, and it is so basic that most of arithmetic secretly leans on it. So it comes as a real shock — the kind that genuinely stalled nineteenth-century mathematicians for years — that this guarantee can simply fail once you step into the ring of algebraic integers of some number fields.

This isn't a matter of picking unlucky numbers to factor, or of some rare pathological ring nobody actually cares about — it happens in a perfectly ordinary-looking ring of algebraic integers, one number field away from the familiar world of ordinary arithmetic.

The most famous example lives in \mathbb{Z}[\sqrt{-5}], the integers of \mathbb{Q}(\sqrt{-5}). There, the ordinary number 6 factors into irreducibles in two genuinely different ways:

This page is about the crisis that caused, and the elegant fix — due to Ernst Kummer and Richard Dedekind — that rescued unique factorisation by inventing an entirely new kind of mathematical object: the ideal.

Worked example: checking the failure is real

It's worth confirming, number by number, that this isn't a typo. First the arithmetic itself: 2 \cdot 3 = 6, and expanding the other side,

Both sides really do equal 6. Now we need to check that 2, 3, 1+\sqrt{-5}, and 1-\sqrt{-5} are all irreducible — that none of them breaks apart any further. The cleanest tool is the norm N(a + b\sqrt{-5}) = a^2 + 5b^2, which is multiplicative (N(xy) = N(x)N(y)) and always a non-negative ordinary integer:

If 2 = xy with neither factor a unit, then N(x)N(y) = 4 with both norms bigger than 1 — forcing N(x) = N(y) = 2. But a^2 + 5b^2 = 2 has no integer solutions at all, so no element of norm 2 exists — 2 cannot split. The identical argument (norm 3 is likewise unreachable) shows 3 is irreducible, and (norms 2 and 3 both unreachable) shows 1 \pm \sqrt{-5} are irreducible too. Four genuinely unbreakable factors, arranged into two different products of the same number. Unique factorisation, as ordinary numbers know it, is simply false in this ring.

The rescue: factor ideals, not numbers

Kummer's insight, refined by Dedekind into its modern form, was to stop factoring numbers and start factoring ideals. An ideal is a set of ring elements closed under addition and under multiplication by anything in the ring — think of it as "the set of all multiples of a number", but promoted to a first-class object in its own right, one sturdy enough to exist even in places where no single generating number does.

This is a genuinely stronger and cleaner statement than anything true of the numbers themselves. Numbers can betray unique factorisation; ideals, provably, never can. In hindsight, "ideal" is a well-chosen name: these are the ideal numbers the ring was always missing, the ones that make the arithmetic behave the way it always should have.

How it heals the example

Here is the reconciliation, informally. Write \mathfrak{p}_2 = (2,\, 1+\sqrt{-5}), \mathfrak{p}_3 = (3,\, 1+\sqrt{-5}), and \mathfrak{p}_3' = (3,\, 1-\sqrt{-5}) — these turn out to be exactly the prime ideals hiding beneath our four "unbreakable" numbers. At the ideal level:

Multiply out both of our original factorisations of 6 in terms of these prime ideals, and they land on exactly the same product:

The numbers 2, 3, 1+\sqrt{-5}, 1-\sqrt{-5} were never truly the finest atoms of this ring — they were only coarse bundles of the real prime-ideal atoms \mathfrak{p}_2, \mathfrak{p}_3, \mathfrak{p}_3', bundled two different ways by two different numbers. Unique factorisation was never actually lost. We had simply been trying to factor the wrong kind of object.

A concrete look at a prime ideal with no generator

It's worth seeing why \mathfrak{p}_2 = (2,\, 1+\sqrt{-5}) genuinely needs two generators — why no single element of \mathbb{Z}[\sqrt{-5}] produces it alone. By definition,

This set contains 2 itself, and 1+\sqrt{-5}, and therefore also their difference \sqrt{-5} - 1, and sums and multiples of all of these — a whole sub-lattice of the ring. If some single element \pi generated all of \mathfrak{p}_2 by itself, its norm would have to be N(\pi) = 2 (the ideal's "size" relative to the whole ring, matching how (2) = \mathfrak{p}_2^{\,2} has norm 4 = 2^2). But we already showed a^2 + 5b^2 = 2 has no integer solutions — no element of the ring has norm 2 at all. So \mathfrak{p}_2 is a perfectly good, perfectly well-defined ideal — closed under addition, closed under multiplication by the ring — that simply is not "the multiples of any one number". It is a non-principal ideal: exactly the kind of new object that has to exist once numbers alone stop being enough to carry the arithmetic.

Why anyone bothered: Fermat's Last Theorem

This wasn't idle abstraction — it was forced by a very concrete crisis. In the 1840s, Gabriel Lamé announced a proof of Fermat's Last Theorem by factoring x^p + y^p using p-th roots of unity — but his argument silently assumed unique factorisation in the resulting ring of integers, exactly the assumption we've just seen can fail. Kummer spotted the gap, and rather than abandon the approach, he built the "ideal numbers" that repair it. With them he proved Fermat's Last Theorem for every regular prime exponent (all primes below 100 except 37, 59, and 67) — a landmark result that stood as the best partial progress on the theorem for over a century, until Wiles's full proof in 1994. A patch for a factorisation problem became, almost incidentally, the deepest attack yet on one of the oldest open problems in mathematics.

Kummer's own fix was surprisingly indirect: rather than defining new objects outright, he introduced formal "ideal numbers" via the divisibility conditions they would have to satisfy, a clever but slightly mysterious workaround. It was Dedekind, about twenty years later, who saw how to make the idea fully rigorous and concrete: instead of an ideal number that might not "really" exist, use the actual set of ring elements it would divide — precisely the sets \mathfrak{p}_2, \mathfrak{p}_3, \mathfrak{p}_3' we computed with above. That reformulation — trade a mysterious formal symbol for a concrete, checkable set — is the "ideal" as every mathematician still uses the word today. (A rival approach, Leopold Kronecker's theory of "divisors", solved the same problem differently and independently; Dedekind's set-theoretic version is the one that stuck.)