Germany's Chancellor Angela Merkel on Saturday said the state must rigorously fight right-wing extremism following the murder of a prominent politician.

The arrest of a man with suspected far-right sympathies over the shooting this month of Walter Luebcke, a regional ally of Merkel known for his pro-migrant views, shocked Germany and prompted calls for a more pro-active government response to anti-immigrant extremists.

Merkel, speaking in Dortmund at an annual gathering of Protestant churches, said that right-wing extremism must be fought "without any taboo".

"Otherwise we will have a complete loss of credibility," she said, adding that the government took the issue "very, very seriously". Her Interior Minister, Horst Seehofer, made similar comments last week.

Germany is home to some 12,700 potentially violent far-right radicals, domestic intelligence agency BfV estimates, and a Civey poll showed 60 percent of Germans think the government is doing too little to oppose them.

Luebcke, the head of the district government in Kassel in the state of Hesse, was shot in the head at close range on the terrace of his home. A 45-year-old named by police as Stephan E. was arrested at the weekend after they found his DNA at the scene.

Investigators said he was a known right-wing radical in the 1980s and 1990s.