In the academic year 1917/18 Hasse was stationed at Kiel on his naval duties and he was able to attend the lectures of Otto Toeplitz. On leaving the navy he entered the University of Göttingen. His teachers there included Edmund Landau, Hilbert, Emmy Noether and Hecke. In fact he was most influenced by Hecke despite the fact that Hecke left Göttingen to take up an appointment in Hamburg only a few months after Hasse arrived in Göttingen.
It might be supposed that Hasse would have followed Hecke to Hamburg but he did not take this route, going to study under Hensel at Marburg in 1920. Hensel's work on p-adic numbers was to have a major influence on the direction of Hasse's research. In October 1920 Hasse discovered the 'local-global' principle which shows that a quadratic form that represents 0 non-trivially over the p-adic numbers for each prime p, and over the real numbers, represents 0 non-trivially over the rationals. The importance of this result, now known as the Hasse principle, is that both the representability of a number by a given form and whether two forms are equivalent can be decided using only local information.
The 'local-global' principle and its applications form an important part of his doctoral thesis Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen Ⓣ of 1921 and also of his habilitation thesis Über die Aquivalenz quadratischer Formen im Körper der rationalen Zahlen Ⓣ.
In 1922 Hasse was appointed a lecturer at the University of Kiel, then three years later he was appointed professor at Halle. During his time at Kiel, Hasse kept in close contact with the mathematicians at Hamburg including Artin, Hecke, Ostrowski and Schreier. He extended Heinrich Weber's work on class field theory writing several important papers and starting work on his famous report on class field theory which included the contributions of Kronecker, Heinrich Weber, Hilbert, Furtwängler and Takagi. As H M Edwards says in :-
... [the text], like any good exposition, contained a great deal of Hasse's own reworking of the material.At Halle Hasse obtained fundamental results on the structure of central simple algebras over local fields. In 1930 Hensel retired from Marburg and Hasse was appointed to fill his chair. While in Marburg he began joint work with Brauer and Emmy Noether on simple algebras, culminating in the complete determination of what is today called the Brauer group of an algebraic number field. He also started work on elliptic curves and, with Baer, on topological fields. At this time he obtained a result that is particularly associated with his name, when (inspired by Mordell and Davenport) he proved the analogue of the Riemann Hypothesis for zeta functions of elliptic curves.
The year 1933 was to be significant for all of Germany and for Hasse in particular. In that year the Nazis came to power and its effect on mathematics in Germany was profound. Mac Lane, who was at Göttingen at this time writes in :-
On April 7, 1933, a new law ... summarily dismissed all those who were Jewish ...The effect on the Mathematical Institute [at Göttingen] was drastic. ... All told, in 1933 eighteen mathematicians left or were driven out from the faculty at the Mathematical Institute in Göttingen.When Weyl resigned from his professorship in Göttingen, Hasse received an offer as his successor. As Edwards says in :-
... [Hasse] appeared to be potentially acceptable to the Nazis and yet was a mathematician of the highest calibre.Following the advice of Weyl, Hasse decided to accept the offer, although (see  and ) he met with fierce opposition from the fundamentalist Nazi functionaries within the Mathematics Institute and the University. In 1934 he was appointed to Göttingen.
It is hard to understand exactly what Hasse's views were in the middle of this political mess. Edwards sums up the position in  as he saw it:-
Hasse's political views and his relations with the Nazi government are not easily categorised. On the one hand, his relations with his teacher Hensel, who was unambiguously Jewish by Nazi standards, were extremely close, right up to Hensel's death in 1941 ... One of his most important papers was a collaboration with Emmy Noether and Richard Brauer, both Jewish, published in 1932 in honour of Hensel's 70th birthday. ... Hasse did not compromise his mathematics for political reasons ... he struggled against Nazi functionaries who tried (sometimes successfully) to subvert mathematics to political doctrine ... On the other hand, he made no secret of his strong nationalistic views and his approval of many of Hitler's policies.At one point, Hasse hoped to increase his influence with the authorities and so he applied for membership in the Nazi Party. But one of Hasse's antecendents was a Jew and, therefore, membership was not granted. Officially his application was put on hold till after the war. See  and .
From 1939 until 1945 Hasse was on war leave from Göttingen and he returned to naval duty, working in Berlin on problems in ballistics. He returned to Göttingen where, in September 1945 he was dismissed from his post by the British occupation forces. His right to teach was terminated and he refused the offer of a research only position, moving to Berlin in 1946 when he took up a research post at Berlin Academy. H W Leopoldt writes in :-
When he resumed teaching in 1948 in Berlin he attracted a large audience. In his first official lecture he compared aesthetic principles working in music and in number theory. Most of his examples regarding music he took from the late piano sonatas of Beethoven, which he - always an ardent piano player - intimately studied during those years. I had just begun to study mathematics, and this lecture made a lasting impression on me, in fact, it decided the further course of my studies.In May 1949, Hasse was appointed professor at the Humboldt University in East Berlin. His work on determining arithmetical properties of abelian number fields Über die Klassenzahl abelscher Zahlkörper Ⓣ was published at this time. At about the same time his textbook Zahlentheorie Ⓣ appeared which contained, for the first time, a systematic introduction to algebraic number theory based on the local method. It was later translated into English.
In 1950 Hasse was appointed to Hamburg where he continued to teach until he retired in 1966. His influence is summed up in  where Edwards writes:-
One of the most important mathematicians of the twentieth century, Helmut Hasse was a man whose accomplishments spanned research, mathematical exposition, teaching and editorial work.This reference to editorial work is made because Hasse was editor of Crelle's Journal for 50 years.
Hasse was honoured by many organisations during his life. Among them were:
Deutsche Akademie der Naturforscher Leopoldina zu Halle,
Akademie der Wissenschaften zu Göttingen,
Finnische Akademie der Wissenschaften in Helsinki,
Deutsche Akademie der Wissenschaften zu Berlin,
Akademie der Wissenschaften und der Literatur in Mainz,
Deutscher Nationalpreis 1.Klasse für Wissenschaft und Technik,
Spanische Akademie der Wissenschaften zu Madrid,
Doctor honoris causa Universität Kiel,
Cothenius Medaille der Deutschen Akademie der Naturforscher Leopoldina.
Article by: J J O'Connor and E F Robertson