Denmark’s Prime Minister Mette Frederiksen says she is “annoyed and surprised” after Donald Trump cancelled his visit over the future of Greenland.

“Our preparations were well under way,” she said, adding she had been looking forward to meeting the US president.

Ms Frederiksen reiterated that the island of Greenland, an autonomous Danish territory, was “not for sale”.

Mr Trump was scheduled to visit on 2 September at the invitation of Denmark’s Queen Margrethe II.

But last week, he suggested that his administration was interested in buying Greenland – an idea that Ms Frederiksen at the time described as “absurd”.

What has the Danish PM said now?

In a statement to reporters on Wednesday, Ms Frederiksen said the potential sale of Greenland had “clearly been rejected” by the island’s leader, Kim Kielsen, which was “a position I share of course”.

She said despite this, the US was one of Denmark’s closest allies, and that the president’s two-day visit would have been an “opportunity to celebrate Denmark’s close relationship to the US”.

“This does not change the character of our good relations and we will continue our dialogue on how we can deal with challenges we are facing,” she said, adding that her country’s invitation to visit “remains open”.

“Developments in the Arctic call for further co-operation with the US, Denmark, Greenland and the Faroe Islands, and I’d like to underline that still stands,” Ms Frederiksen said.

• Donald Trump and Greenland: Why would he want to buy it?

How did we get here?

While praising Denmark as a “very special country”, Mr Trump said in a tweet on Tuesday that his planned visit would no longer go ahead because Ms Frederiksen had “no interest in discussing the purchase of Greenland”.

• Greenland: What you need to know

Mr Trump had earlier confirmed reports that he was interested in buying Greenland. When asked on Sunday if he would consider trading a US territory for the island, he replied: “Well, a lot of things could be done.”

“Essentially it’s a large real estate deal,” he said.

Then, on Monday, the US president posted an image on social media showing a tall golden skyscraper among the homes of a small village on the island.

How have Danes reacted?

The cancellation of what was considered an important state visit in Denmark was described as a “farce” by the leader of the populist Danish People’s Party, Kristian Thulesen Dahl.

“What is this man thinking of though? And with grounds that are worthy of an April Fools’ joke,” he wrote in a tweet.

Danish Conservative MP Rasmus Jarlov accused Mr Trump of lacking respect for his country in a tweet on Wednesday.

Former foreign minister Kristian Jensen said Mr Trump’s move had resulted in “total chaos”.

“It has gone from a great opportunity for enhanced dialogue between allies to a diplomatic crisis,” he said, adding: “Need to get the co-operation back on track.”

A spokeswoman for the leftist Red-Green Alliance, Pernille Skipper, said: “Trump lives on another planet. Self-sufficient and disrespectful.”

Pia Kjaersgaard, the populist former speaker of the Danish parliament, said it showed a “complete lack of respect”, adding: “Rude behaviour to the Danish people and the Queen, who invited him.”

Danish royal expert and historian Lars Hovbakke Sorensen said the whole episode was “very strange”.

“This is a violation of all traditions and etiquettes and makes it seem like the Trump administration views state visits in a completely different way,” he told Danish broadcaster TV2.

Why would Greenland appeal to Trump?

Mr Trump has reportedly taken an interest in Greenland, in part, because of its natural resources, such as coal, zinc, copper and iron ore.

But while Greenland might be rich in minerals, it currently relies on Denmark for two thirds of its budget revenue. It has high rates of suicide, alcoholism and unemployment.

Two people briefed on the discussions told the New York Times the president was also interested in Greenland’s “national security value” because of its location.

The US has long seen the island, which sits along a direct route from Europe to North America, as being strategically important. It established the Thule air force and radar base there at the start of the Cold War, which now covers space surveillance and forms the northernmost part of the US ballistic missile early warning system.

Meanwhile, new Arctic sea routes are opening up as climate change continues to accelerate the melting of ice in the region.

• Is the Arctic set to become a main shipping route?

Mr Trump’s latest bid also comes at a time that China is taking a big interest in the area. Last year, a Chinese state-owned construction company announced plans to build new airports in Greenland – but withdrew them in June this year.

Republican Representative Mike Gallagher described Mr Trump’s idea as a “smart geopolitical move”.

“The United States has a compelling strategic interest in Greenland, and this should absolutely be on the table,” he tweeted.

Where exactly is Greenland?

Greenland is the largest island in the world – after Australia, which is defined as a continent in its own right.

It is an autonomous Danish territory, located between the North Atlantic and Arctic oceans.

It has a population of about 56,000 people concentrated around the coastline. Almost 90% are indigenous Greenlandic Inuit people. It has a limited self-government and its own parliament.

• The stark photo highlighting Greenland’s ice loss
• ‘Why I translate all of Trump’s tweets into Chinese’

More than 80% of the island is covered by an ice cap which is feared to be melting due to global warming. The ice melt has increased access to the island’s mineral resources.

But it is also believed that the receding ice may expose toxic nuclear waste that was left at several US military sites during the Cold War.