Bauer published two important papers in 1903. These were Über einen Satz von Kronecker Ⓣ and Über zusammengesetzte Körper Ⓣ. These papers made an important contribution to Kronecker's question concerning characterising number fields by the splitting behaviour of their primes. Szamuely writes :-
Kronecker called this a 'boundary value problem' (Randwertproblem) because of a (vague) analogy with Cauchy's theorem computing the values of an analytic function on a disc from its values taken at the boundary.What Bauer proved was that if two finite Galois extensions of the rationals have the property that, with at most a finite number of exceptions, the same primes split completely in both extensions, then the two extensions are equal.
In 1909 Bauer was appointed as a privatdozent at the Technical University of Budapest having submitted a habilitation thesis on analytic number theory and function theory. In 1916 he published three further significant papers, namely Zur Theorie der arithmetischen Progression, Zur Theorie der algebraischen Zahlkörper Ⓣ, and Über zusammengesetzte Zahlkörper Ⓣ. Given a finite extension K of the rationals he considered the set P(K) of primes unramified in K and having a prime divisor of degree 1. In the first of the above three papers he showed that the finite extension K is Galois if and only if the primes in P(K) split completely in K. In the second of the papers he completely solved Kronecker's question concerning characterising number fields by the splitting behaviour of their primes. In the third of the papers we listed above :-
Bauer pursued the study of the topic, obtaining more general results about the decomposition of (not necessarily split) prime ideals in composita of finite Galois extensions.He was appointed as an extraordinary professor on 18 June 1918 and he was officially inducted to that role on 5 November 1918 by József Kürschák. Kürschák was himself a mathematician but it was as rector of the university that welcomed Bauer.
The Eötvös Lóránd Mathematical and Physical Society was originally called the Mathematical and Physical Society but was renamed in honour of its first president after his death in 1919. The Society created the Gyula König Prize, named after König who had died in 1913. It was fitting that Bauer, who was a student of König, should be the first recipient of the prize in 1922. His career up to this point sounds like one of achievement which was being recognised. Indeed it was to some extent recognised but his status within the university remained at a level below which would be expected for someone of his talents. Rédei writes  (as translated by Tamás Szamuely in ) that, in 1924, Bauer :-
... obtained a position as a secondary school teacher while maintaining his service at the university. The positive effect of this was that he was not a temporary employee any more, and was entitled to a pension. On the other hand around this time unfortunate circumstances arose that cast a shadow on his remaining life. He started being insulted because of his [Jewish] origin, which for the moment meant the aggressive disturbance of his lectures by some of his students at the Technical University. This was all the more undeserved as he prepared his lectures with extreme care, and he was a model of a teacher sacrificing himself for the benefit of his pupils. The behaviour of his students was in line with the political tendencies of the period, and was one of the reasons why he was forced to take early retirement (around 1936). The insults he had to suffer caused much long-lasting pain to Bauer.Bauer had other problems to contend with in addition to the disgraceful anti-Semitic behaviour of his students. He was also beginning to lose his sight which naturally caused him considerable anxiety. In fact his eyesight stabilised and over the last years of his life there was very little further deterioration. The anti-Semitism, however, became worse during World War II. Horthy, who headed the Hungarian government, disliked Adolf Hitler but was opposed to the communist forces to the east. He tried to steer a middle course with Germany but in March 1944 Hitler offered Horthy a choice between full cooperation with German policies or to be treated as an enemy and be occupied by German forces. He chose the former, which led to much more severe anti-Semitic policies being carried out. Bauer celebrated his 70th birthday on 20 September 1944 forced to wear a yellow star and forced to share his home. Rédei writes  (as translated by Tamás Szamuely in ):-
Needless to say, there was no public feast as he was pursued for political reasons, and only very few of his students and colleagues could visit him on the occasion. I expressed my greetings by way of a letter, sending him the proofs of my paper dedicated to him that was to appear in the journal Acta Scientiarium Mathematicarum published in Szeged. In his reply, the last letter I received from him, he expressed his thanks for the gesture of his former student, and complained that he was forced to share his flat with another person and as a consequence could not even regard his writing desk as his own.Horthy tried to make peace with the allies and announced a preliminary deal on 15 October. He was immediately removed from power by the Germans who put Ferenc Szálasi, the leader of the Hungarian fascist party, in control. Immediately he moved his forces against the Hungarian Jews and Bauer was sent to Tattersaal camp. He was able to return to his home but was then forced to the ghetto where he suffered extreme hardship until liberated after the German and Hungarian armies were forced back. It was to be a very short few days of freedom for in February 1944 he was on his way to buy eye drops when he collapsed in the street, hitting his head. He was taken to hospital where he died soon after being admitted.
Article by: J J O'Connor and E F Robertson