Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The proportion of failures of the Hasse norm principle   For any number field we calculate the exact proportion of rational numbers
which are everywhere locally a norm but not globally a norm from the number
field.

18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci