Train is also subsidized. In fact, while air transport benefits from tax exemption, train receives substantial infrastructure subsidy. The following report puts rail subsidy to be around 75B EUR, whereas air within 27B to 35B. The study is from 2007, but I doubt it has changed much.

https://www.eea.europa.eu/publications/technical_report_2007...

Very informative report. But as per its conclusion: "subsidies could be justified to balance the existence of external costs of competing modes, e.g. when an internalisation of the externality, for whatever reason, encounters strong political opposition from vested interests. This applies to some of the subsidies for public transport, especially rail, since its competing modes, particularly road and air, have significantly higher external costs".

In other words, it makes sense to subsidize rail, it doesn't make sense to subsidize modes with higher externalities.

It may very well be that rail subsidy makes sense and air subsidy doesn't. I'm not arguing for or against it. The original question pertains to why air ticket is cheaper than train. You and a number of posters here attributed that to subsidy, and I just wanted to point out that rail transport in Europe receives even larger subsidy than air transport.

There are many factors affecting pricing, for example occupancy rate. Load factor of aircrafts are in the 80 percentage - [0], whereas it's often under 60% for long distance Deutsche Bahn's trains in Germany - [1]

Another possibility is that train operators are often state owned and therefore face less competition and are less efficient than airliners, who are used to operate in a cutthroat environment.

Note that I don't claim either of the above cause train's pricing disadvantage, merely they are factors to consider. OTOTH attributing pricing advantage to subsidy is a lazy take of a complex issue.

[0] - https://www.statista.com/statistics/658830/passenger-load-fa...

[1] - https://www.statista.com/statistics/654973/long-distance-tra...