Hasse Norm Principle

• Hasse norm principle is a local–global assertion in finite field extensions ensuring that elements are global norms when local norm conditions are met and the knot group vanishes.
• It connects algebraic number theory with cohomological methods, interpreting the obstruction via the Tate–Shafarevich group of norm one tori and Galois cohomology.
• It underpins explicit group-theoretic criteria and statistical results that classify and quantify the failure of local-to-global norm equations.

The Hasse norm principle is the local–global assertion that for a finite extension of global fields $K/k$, every element of $k^\times$ that is a norm in every completion is already a global norm. In idelic form, it is the equality

$$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$

and its failure is measured by the finite quotient

$$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$

often called the knot group. The classical theorem of Hasse proves the principle for cyclic extensions, while modern work recasts the obstruction in terms of norm one tori, Galois cohomology, decomposition groups, and arithmetic statistics (Browning et al., 2014, Hoshi et al., 2019).

1. Classical formulation and obstruction group

For a finite extension of global fields $L/K$, the field norm

$N_{L/K}:L^\times\to K^\times$

and the idelic norm

$N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$

define two nested classes of elements of $K^\times$: global norms $N_{L/K}(L^\times)$, and everywhere local norms $K^\times\cap N_{L/K}(\mathbb A_L^\times)$. The Hasse norm principle asserts that these coincide. Equivalently, if $k^\times$0 lies in $k^\times$1 for every place $k^\times$2 of $k^\times$3, then $k^\times$4 (Browning et al., 2014).

The obstruction is the knot group

$k^\times$5

The principle holds if and only if $k^\times$6, or equivalently $k^\times$7. In the literature summarized here, this quotient is the standard arithmetic measure of local–global failure, and it is also the form in which quantitative questions are posed (Browning et al., 2014).

The classical benchmark is Hasse’s theorem: if $k^\times$8 is cyclic, then the Hasse norm principle holds. For finite abelian groups, the existence of failures is completely characterized: there exists a $k^\times$9-extension of a number field failing the Hasse norm principle if and only if $$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$0 is non-cyclic (Frei et al., 2015). In particular, cyclicity eliminates the obstruction, but non-cyclicity does not determine its size or frequency.

A recurrent misunderstanding is to identify the Hasse norm principle with arbitrary local–global statements for norm-form equations. The norm equation $$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$1 is the direct torsor-theoretic avatar of the principle, but many nearby problems involve families of norm equations with a varying right-hand side and are therefore adjacent rather than identical to the classical Hasse norm problem (Browning et al., 2011).

2. Cohomological and toric interpretation

The central geometric object is the norm one torus

$$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$2

Ono’s theorem identifies the Hasse norm obstruction with the Tate–Shafarevich group of this torus: $$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$3 Thus $$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$4 if and only if the Hasse norm principle holds for $$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$5 (Hoshi et al., 2019).

This identification places the problem in the arithmetic of algebraic tori. For a smooth compactification $$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$6 of $$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$7, Voskresenskii’s exact sequence

$$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$8

links the defect of weak approximation

$$N_{K/k}(K^\times)=k^\times\cap N_{K/k}(\mathbb A_K^\times),$$9

to the Hasse norm obstruction. Consequently, $$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$0 implies both weak approximation for $$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$1 and the Hasse norm principle for $$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$2 (Hoshi et al., 2019, Rivera-Mesas, 2021).

For a Galois splitting field $$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$3 with $$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$4, the character lattice $$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$5 becomes a $$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$6-lattice, and non-Galois norm one tori are described by the Chevalley module $$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$7, where $$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$8. It is defined by

$$\mathfrak K(K/k)=\frac{k^\times\cap N_{K/k}(\mathbb A_K^\times)}{N_{K/k}(K^\times)},$$9

Flasque resolutions of $L/K$0 reduce the calculation of $L/K$1 to finite-group cohomology, and this is the mechanism behind explicit classifications in small degree and for specific families of transitive groups (Hoshi et al., 2019, Macedo et al., 2019).

In the Galois case, Tate’s description makes the local–global structure especially transparent: $L/K$2 and dually

$L/K$3

Here the decomposition groups $L/K$4 control whether the global cohomology class survives as a Hasse norm obstruction or is detected locally and reappears as a weak-approximation defect (Rivera-Mesas, 2021).

3. Explicit criteria and group-theoretic families

Beyond cyclic extensions, the modern theory is dominated by precise group-theoretic criteria. For Galois dihedral extensions with group $L/K$5 of order $L/K$6, the pattern is completely explicit. If $L/K$7 is odd, then the Hasse norm principle and weak approximation for the norm one torus both hold. If $L/K$8 is even, then the obstruction is either trivial or $L/K$9: HNP holds exactly when some local decomposition group contains a Klein four subgroup, and otherwise HNP fails while weak approximation holds (Rivera-Mesas, 2021).

For alternating closures, the picture is uniformly positive. If $N_{L/K}:L^\times\to K^\times$0 is a degree $N_{L/K}:L^\times\to K^\times$1 extension of number fields with normal closure $N_{L/K}:L^\times\to K^\times$2 satisfying

$N_{L/K}:L^\times\to K^\times$3

then the Hasse norm principle holds for $N_{L/K}:L^\times\to K^\times$4, and weak approximation holds for the norm one torus $N_{L/K}:L^\times\to K^\times$5 (Macedo, 2018). This sharply contrasts with the quartic $N_{L/K}:L^\times\to K^\times$6-case, where failure can occur.

A broad sufficient criterion is available for metacyclic closures. If the Galois closure group $N_{L/K}:L^\times\to K^\times$7 is metacyclic and has trivial Schur multiplier

$N_{L/K}:L^\times\to K^\times$8

then the Hasse norm principle holds for $N_{L/K}:L^\times\to K^\times$9. In that setting the relevant obstruction injects into $N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$0, so triviality of the multiplier kills the Hasse norm obstruction. The same argument yields the Tamagawa number formula

$N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$1

for the associated norm one torus (Hoshi et al., 18 Mar 2025).

The prime-power and prime-square cases admit even more precise descriptions. For Heisenberg extensions of degree $N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$2 or $N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$3, with Galois closure group the extraspecial group $N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$4, the obstruction group can be $N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$5, $N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$6, or $N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$7, and the criterion is expressed in terms of whether decomposition groups contain suitable noncyclic subgroups, especially $N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$8 (Hoshi et al., 19 Mar 2025). For finite separable extensions of odd prime-squared degree $N_{L/K}:\mathbb A_L^\times\to \mathbb A_K^\times$9, failure can occur only when

$K^\times$0

for some $K^\times$1, and then

$K^\times$2

This recovers the earlier theorem of Drakokhrust–Platonov on adequate extensions of prime-squared degree (Oki, 14 Aug 2025).

Explicit computational classifications extend much further. For norm one tori of degree $K^\times$3, $K^\times$4, $K^\times$5 has been determined for every transitive Galois closure group $K^\times$6, and the Hasse norm principle is then decided by concrete decomposition-group criteria involving subgroups such as $K^\times$7, $K^\times$8, $K^\times$9, or $N_{L/K}(L^\times)$0 (Hoshi et al., 2019). Sporadic simple groups have also entered the subject: for $N_{L/K}(L^\times)$1- and $N_{L/K}(L^\times)$2-closures, $N_{L/K}(L^\times)$3 is either $N_{L/K}(L^\times)$4 or $N_{L/K}(L^\times)$5, and in the exceptional cases HNP is decided by the existence of a place whose decomposition group contains $N_{L/K}(L^\times)$6, $N_{L/K}(L^\times)$7, $N_{L/K}(L^\times)$8, or $N_{L/K}(L^\times)$9, depending on the subgroup $K^\times\cap N_{L/K}(\mathbb A_L^\times)$0 (Hoshi et al., 2022).

The non-Galois square-free setting is also known to exhibit genuine failures. For any square-free composite integer $K^\times\cap N_{L/K}(\mathbb A_L^\times)$1 divisible by at least one of $K^\times\cap N_{L/K}(\mathbb A_L^\times)$2, $K^\times\cap N_{L/K}(\mathbb A_L^\times)$3, $K^\times\cap N_{L/K}(\mathbb A_L^\times)$4, or $K^\times\cap N_{L/K}(\mathbb A_L^\times)$5, there exists a finite extension of degree $K^\times\cap N_{L/K}(\mathbb A_L^\times)$6 for which the Hasse norm principle fails (2307.12550).

4. Quantitative and statistical aspects

A second major direction asks not whether HNP can fail, but how often it fails. For a fixed number field $K^\times\cap N_{L/K}(\mathbb A_L^\times)$7, Browning and Newton compute the exact proportion of rational numbers that are everywhere locally norms but not global norms. If

$K^\times\cap N_{L/K}(\mathbb A_L^\times)$8

$K^\times\cap N_{L/K}(\mathbb A_L^\times)$9

and $k^\times$00, then

$k^\times$01

where $k^\times$02 (Browning et al., 2014). Thus the finite obstruction group governs not only solvability, but also the asymptotic frequency of local–global failure among local norms.

Over global function fields the same phenomenon persists. For a finite extension $k^\times$03, with $k^\times$04 and full constant field $k^\times$05, one has

$k^\times$06

Equivalently, the asymptotic proportion of polynomials that are local-but-not-global norms is

$k^\times$07

The function-field argument must also accommodate inseparability and constant-field phenomena (Mânzăţeanu et al., 2020).

A different statistical problem orders the extensions themselves by discriminant. For biquadratic extensions of $k^\times$08, the number $k^\times$09 of fields with $k^\times$10 satisfies

$k^\times$11

whereas the number $k^\times$12 of such fields failing HNP satisfies

$k^\times$13

Hence

$k^\times$14

Failures are infinite in number but zero-density in the discriminant-ordered biquadratic family (Rome, 2017).

For general finite abelian groups, two complementary discriminant-ordered results are now available. One theorem shows that for

$k^\times$15

where $k^\times$16 is the smallest prime dividing $k^\times$17, $k^\times$18 of $k^\times$19-extensions fail HNP, whereas for all other nontrivial finite abelian groups a positive proportion fail (Frei et al., 2015). A later refinement proves that for every nontrivial finite abelian group $k^\times$20, the density of $k^\times$21-extensions satisfying HNP exists; it equals $k^\times$22 if $k^\times$23 is cyclic, where $k^\times$24 is the smallest prime dividing $k^\times$25, and it lies in $k^\times$26 otherwise (Koymans et al., 2023). These results show that abelian HNP failure is neither purely sporadic nor generically dominant; its density is group-theoretically stratified.

5. Generalizations, variants, and adjacent problems

The multinorm problem replaces a single norm equation by

$k^\times$27

or equivalently $k^\times$28 for the étale algebra $k^\times$29. In the presence of a cyclic factor $k^\times$30, Bayer-Fluckiger, Lee, and Parimala construct an explicit finite obstruction group $k^\times$31 and a Brauer–Manin map

$k^\times$32

such that

$k^\times$33

This recovers the classical Hasse norm theorem in the one-factor case and extends it to multinorm tori and metacyclic settings (Bayer-Fluckiger et al., 2015).

A projective variant asks not for a single scalar norm, but for the existence of a norm point on a projective line. For separable extensions $k^\times$34, the projective Hasse norm principle is formulated as

$k^\times$35

or equivalently via the projective norm map

$k^\times$36

In general this projective principle is independent from the conjunction of the ordinary Hasse norm principles of the constituent fields, but for independent Galois extensions it follows from them; in particular, cyclic extensions satisfy the projective analogue of Hasse’s theorem (Rüd et al., 2024).

Norm-form varieties give a closely related but distinct class of local–global problems. For an irreducible quadratic polynomial $k^\times$37 and a quartic extension $k^\times$38 containing a root of $k^\times$39, the affine variety

$k^\times$40

satisfies the Hasse principle and weak approximation. The proof uses analytic methods and the classical Hasse norm principle only for the auxiliary quadratic field $k^\times$41, not for the quartic extension $k^\times$42. Accordingly, this result belongs to the broader study of norm-form varieties rather than constituting a direct theorem about the classical Hasse norm principle for $k^\times$43 (Browning et al., 2011).

6. Arithmetic-topological analogue

Arithmetic topology provides a topological analogue of the Hasse norm principle. In Morishita’s dictionary, a closed oriented $k^\times$44-manifold $k^\times$45 plays the role of the ring of integers of a number field, knots correspond to primes, links to finite sets of places, and finite branched coverings to field extensions. Within this framework, the topological idele group of a pair $k^\times$46 with $k^\times$47 a very admissible link is

$k^\times$48

and the principal ideles are defined by a diagonal map

$k^\times$49

For a finite cyclic branched covering

$k^\times$50

of an integral homology $k^\times$51-sphere $k^\times$52, the pushforward

$k^\times$53

plays the role of the norm map. The topological Hasse norm theorem then states

$k^\times$54

This is the exact analogue of

$k^\times$55

for finite cyclic extensions of number fields (Tashiro, 2024).

The proof uses explicit Seifert-surface formulas for principal topological ideles, quotient identifications

$k^\times$56

and an exact sequence for cyclic coverings analogous to Galois descent. The result is not the classical theorem itself, but it isolates the same structural ingredients: global principality, local norm conditions, cyclic descent, and a local–global passage. It therefore shows that the conceptual form of the Hasse norm principle survives outside algebraic number fields, in a $k^\times$57-manifold setting governed by topological class field theory (Tashiro, 2024).