There can hardly be any more extraordinary story from the Hollywood golden age than that of Hedy Lamarr; a very beautiful star with a moderate acting talent but an untutored brilliance in science and engineering that should by now be getting her compared to Nikola Tesla, or maybe even a neglected female scientist like Rosalind Franklin. Her tragedy was that she was in the wrong business, precisely that business that promotes beauty over brains – the movie business.

Alexandra Dean’s excellent and important documentary about her is very instructive – a parable of modern sexual politics and assumptions about science. Even now, many can’t believe in their hearts that movie star Hedy Lamarr really was a scientist, or scientist manqué. The accomplishment simply doesn’t square with the accepted female star biography narrative into which Lamarr otherwise fits: movies, husbands, poignant reclusive decline etc. Many film encyclopedias and reference books simply omit what was important about Hedy Lamarr.

As a Jewish refugee from Hitler’s Austria and already a celebrity on account of her nude appearance as a teenager in the steamy German film Ecstasy, Hedy Kiesler (with her new stage name “Lamarr”) was signed up by Louis B Mayer. It was in the 40s that Lamarr pursued inventing in her spare time after a hard day in the studio. And, without any of the official support and funding that was lavished on a battalion of unproductive dunderheads and with some hobbyist equipment given to her by Howard Hughes, she contributed to a discovery that was to change the course of the 20th century.

With her friend, composer George Antheil, she submitted a patent for a secure communications “frequency hopping” technology that the US Navy was to use after the patent’s lapse without paying her (or him) and that was to become the basis for Bluetooth and wifi. Lamarr was to die in near poverty, while her invention was worth billions. Finally, due to some gallant investigative reporting from Forbes magazine journalist Fleming Meeks in the 90s, she was given some recognition. But no money.

What more might this remarkable woman have achieved if she had only got some support? Well, Dean shows that she had quite a few achievements to her name. She invented the ski resort of Aspen, Colorado, by noticing this place’s resemblance to Austria and the great skiing possibilities – she bought a chalet there and sowed the seeds for what it was to become. Lamarr also, arguably, invented modern cosmetic surgery in her decline by telling the doctors where and how incisions should be made.

She always had a certain worldly style and feline sexiness, and a sprightly cosmopolitan elegance that she had brought over from the Freud/Zweig-era Vienna of her girlhood. It entranced a generation of male moviegoers, though was never quite in the Dietrich or Garbo class.

Lamarr knew that the movies she was being cast in were mostly pretty forgettable. So she started co-producing her own films – again, an almost unheard-of innovation in that male mogul world – including the madly ambitious but financially ruinous Loves of Three Queens (1954) in which she played Helen of Troy, Empress Josephine and medieval myth-heroine Genevieve of Brabant.

She never had any serious prizes to her name, although she was recognised by something called the Golden Apple awards, which, incredibly, were to recognise “good behaviour” in entertainers and which actually ran until 2001. In 1949, Hedy Lamarr won its “Sour Apple” prize for least cooperative actress.

And all the time she was pursuing inventions, suggesting aircraft design modifications to Hughes and enduring multiple marriages and divorces, all of which ended badly – especially when, already unbalanced by the “speed” pills the studio had required her to take for the work schedule and a somewhat imperious manner, she bizarrely sent her on-set body double to testify in the divorce court on her behalf. The judge was not impressed.

The money and the work dried up. Lucille Ball – to whom she did have a faint facial resemblance – mocked her cruelly on TV. The film shows us a stilted TV appearance on The Merv Griffin Show, with Woody Allen, to promote a ghosted autobiography the very existence of which she appears to disdain.

Lamarr was an enigma: a great brain trapped in a silly, spurious image of glamour, while her real talent was allowed to wither. A sad but fascinating story.