Leopoldt’s conjecture

Multiplication can be reversible without leaving an integer-like number system. For example, $$(1+\sqrt{2})(\sqrt{2}-1)=1.$$ Both factors are whole-number combinations of $\sqrt{2}$. Such a number and its inverse are units. In general, a number field has a corresponding ring of algebraic integers, and units must have inverses in that ring too. Independent units give different directions for forming products and powers.

Leopoldt asks whether those directions remain independent when measured using prime-based arithmetic. For the prime 3, 82 is closer to 1 than 4 is: their differences are 81 and 3, and 81 contains more factors of 3. This p-adic notion of closeness is different from ordinary distance. Could its measurements introduce extra linear relations among the units? The conjecture says no, for every number field and every prime.[1] [2]

A number system can have several independent ways to multiply by a unit and undo that multiplication. Could prime-based arithmetic introduce new linear relations between those ways? Leopoldt's conjecture says it cannot, for any number field and any prime.

For a concrete unit, use $K=\mathbb Q(\sqrt{2})$. Its algebraic integers include $1+\sqrt{2}$ and its inverse $\sqrt{2}-1$, since their product is 1. These generate reversible multiplication inside the integer ring. Independence of units is about products of whole-number powers, ignoring roots of unity, which have finite order.

The unit theorem gives the number of independent directions: $$r=r_1+r_2-1.$$ Here $r_1$ counts real embeddings of K, and $r_2$ counts conjugate pairs of non-real complex embeddings: arithmetic-preserving ways to place K among the real or complex numbers. For the displayed real quadratic field, $r_1=2$ and $r_2=0$, so r is 1. This example lies in a known family; the question is universal.

For a prime p, closeness is measured by divisibility by powers of p: $|81|_3=1/81$, while $|3|_3=1/3$. A number field can have several such measurements associated with the same p; all are used here. Completing under these distances gives the local fields at the places above p. Use p-adic logarithms to turn unit multiplication into addition. The conjecture requires the measurements of r independent units to have rank r over $\mathbb Q_p$, so no extra p-adic linear dependencies occur. The rank lost, if any, is the Leopoldt defect; the target is defect zero.[1] [2]

No general answer covers every number field and prime.[1]

The March 2026 paper supplies specified field families and transfer criteria using representation theory and explicit units. Those results do not cover arbitrary K and p.[1]

The question also controls how many independent infinite towers of larger number fields can grow in a special way: each step has prime degree, and a single symmetry generates every level's symmetries over the starting field. These towers organize arithmetic across related number systems, instead of one field at a time. A statement about reversible multiplication is therefore connected to the possible ways whole families of number systems can grow.[3]

Local measurements of units also control global extension towers. A $\mathbb Z_p$-extension is an infinite tower whose level-n field is a cyclic extension of degree $p^n$ over the starting field. Leopoldt is equivalent to saying that a number field has exactly $r_2+1$ independent such towers, with $r_2$ the number of pairs of complex embeddings.

This gives a concrete reason for its role in Iwasawa theory: that theory follows arithmetic as one climbs these towers, and the conjecture specifies how many independent directions for the towers exist. The connection reaches beyond checking whether a list of units retains its rank.[3]