mathematics

Hasse norm theorem

Hasse norm theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hasse norm theorem rather than just read about it. In short: In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm. Here to be a global norm means to be an element k of K such that there is an element l of L with N L / K ( l ) = k {\displaystyle \mathbf {N} _{L/K}(l)=k} ; in other words k is a relative norm of some element of the extension field L.

Key takeaways

Hasse norm theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hasse norm theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hasse norm theorem from memory before moving on to harder problems.

Reference excerpt

In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm. Here to be a global norm means to be an element k of K such that there is an element l of L with N L / K ( l ) = k {\displaystyle \mathbf {N} _{L/K}(l)=k} ; in other words k is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then k is a norm from LP; here the "prime" p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean. The theorem is no longer true in general if the extension is abelian but not cyclic. Hasse gave the counterexample that 3 is a local norm everywhere for the extension Q ( − 3 , 13 ) / Q {\displaystyle {\mathbf {Q} }({\sqrt {-3}},{\sqrt {13}})/{\mathbf {Q} }} but is not a global norm. Serre and Tate showed that another counterexample is given by the field Q ( 13 , 17 ) / Q {\displaystyle {\mathbf {Q} }({\sqrt {13}},{\sqrt {17}})/{\mathbf {Q} }} where every rational square is a local norm everywhere but 5 2 {\displaystyle 5^{2}} is not a global norm. This is an example of a theorem stating a local-global principle. The full theorem is due to Hasse (1931). The special case when the degree n of the extension is 2 was proved by Hilbert (1897), and the special case when n is prime was proved by Furtwangler in 1902. The Hasse norm theorem can be deduced from the theorem that an element of the Galois cohomology group H2(L/K) is trivial if it is trivial locally everywhere, which is in turn equivalent to the deep theorem that the first cohomology of the idele class group vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the group H2(L/K) is isomorphic to the Tate cohomology group H0(L/K) which describes which elements are norms, so for cyclic extensions it becomes Hasse's theorem that an element is a norm if it is a local norm everywhere.

See also Grunwald–Wang theorem, about when an element that is a power everywhere locally is a power.

References Hasse, H. (1931), "Beweis eines Satzes und Wiederlegung einer Vermutung über das allgemeine Normenrestsymbol", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 64–69 H. Hasse, "A history of class field theory", in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory, Academic Press, 1973. Chap.XI. G. Janusz, Algebraic number fields, Academic Press, 1973. Theorem V.4.5, p. 156 Hilbert, David (1897), "Die Theorie der algebraischen Zahlkörper", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 4: 175–546, ISSN 0012-0456

Worked examples

Example 1 — a first encounter with Hasse norm theorem

Start with the simplest possible case. Write down what Hasse norm theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hasse norm theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hasse norm theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hasse norm theorem

In research: Hasse norm theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hasse norm theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hasse norm theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Class field theory, Theorems in algebraic number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Hasse norm theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Hasse norm theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hasse norm theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hasse norm theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hasse norm theorem in simple terms?

Why does Hasse norm theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hasse norm theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hasse norm theorem.