It will go down in history as the moment one of the last bastions of male dominance fell. A woman has won the world's most prestigious mathematics prize for the first time since the award was established nearly 80 years ago.

Maryam Mirzakhani, an Iranian maths professor at Stanford University in California, was named the first female winner of the Fields Medal – often described as the Nobel prize for mathematics – at a ceremony in Seoul on Wednesday.

The maths community has been abuzz with rumours for months that Mirzakhani was in line to win the prize. To outsiders her work is esoteric, abstract and impenetrable. But to more qualified minds, she has a breathtaking scope, is technically superb and boldly ambitious. She describes the language of maths as full of "beauty and elegance".

The prize, worth 15,000 Canadian dollars (£8,000), is awarded to exceptional talents under the age of 40 once every four years by the International Mathematical Union. Between two and four prizes are announced each time.

Three other researchers were named Fields Medal winners at the same ceremony in South Korea. They included Martin Hairer, a 38-year-old Austrian based at Warwick University in the UK, Manjul Bhargava, a 40-year old Canadian-American at Princeton University in the US and Artur Avila, 35, a Brazilian-French researcher at the Institute of Mathematics of Jussieu in Paris.

There have been 55 Fields medallists since the prize was first awarded in 1936, including this year's winners. The Russian mathematician Grigori Perelman refused the prize in 2006 for his proof of the Poincaré conjecture.

Mirzhakhani, 37, was among a number of women tipped for the prize in recent years and her success won immediate praise from fellow mathematicians.

Christiane Rousseau, vice president of the International Mathematics Union, said: "It's an extraordinary moment. Marie-Curie had Nobel prizes in physics and chemistry at the beginning of the 20th century, but in mathematics this is the first time we have a woman winning the most prestigious prize there is. This is a celebration for women."

"I am thrilled that this day has finally come," said Sir Tim Gowers, a Fields medallist and mathematician at Cambridge University. "Although women have contributed to mathematics at the highest level for a long time, this fact has not been visible to the general public. I hope that the existence of a female Fields medallist, who will surely be the first of many, will put to bed many myths about women and mathematics, and encourage more young women to think of mathematical research as a possible career."

Born and raised in Iran, Mirzakhani completed a PhD at Harvard in 2004. Her path into mathematics was not a given, though. As a child, her passion was not for numbers, but literature. Her school in Tehran was near a street full of bookshops and because browsing was not allowed, she ended up buying a lot of random books. "I dreamed of becoming a writer," she said in an interview for the Clay Mathematics Institute (CMI) in 2008. "I never thought I would pursue mathematics before my last year in high school."

It was Mirzakhani's brother who first piqued her interest in science. He used to come home from school and talk over what he had learned. He told her the story of the German mathematician, Carl Friedrich Gauss, who displayed his precocious skills as a schoolboy when he worked out in seconds how to sum all the numbers from 1 to 100. (The answer is 5,050 and the trick is to look at pairs that add up to 101.) "That was the first time I enjoyed a beautiful solution, though I couldn't find it myself," she said.

The seed that had been sown began to germinate, with help from her school principal, a strong-willed woman who made every effort to ensure her students had the same opportunities as the boys. As a teenager, Mirzakhani took part in international maths olympiads and won gold medals in 1994 and in 1995. In the first, in Hong Kong, she dropped a single point. At the latter, in Toronto, she finished with a perfect score.

Later, as a student at Sharif University, she befriended inspiring mathematicians and found that the more time she spent on the subject, the more excited she became. Then, at Harvard, she began to work with another Fields medallist, Curt McMullen, and became fascinated with how he made mathematics so simple and elegant.

Most of the problems Mirzakhani works on involve geometric structures on surfaces and their deformations. She has a particular interest in hyperbolic planes, which can look like the edges of curly kale leaves, but may be easier to crochet than explain. According to a citation released by the International Mathematical Union, Mirzakhani won the prize for her "outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces".

Hairer won for his "outstanding contributions to the theory of stochastic partial differential equations", which contain terms that are inherently random. At school, he created audio software that he marketed as "the Swiss army knife of sound editing".

Avila was honoured for his "profound contributions to dynamical systems". Bhargava won for "developing powerful new methods in the geometry of numbers", including elliptic curves used in cryptography.

"The mathematics that has been the most applicable and important to society over the years has been the mathematics that scientists found while searching for beauty; and eventually all beautiful and elegant mathematics tends to find applications," said Bhargava.

Mirzakhani declined an interview, but she told the CMI in 2008 that while maths was not for everyone, many students did not give it a real chance. She did poorly at maths for several years at school because she was not interested in the subject. "I can see that without being excited, mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers," she said.

Speaking to the American Mathematical Society last year, she said the situation for women in maths was still far from ideal. "The social barriers for girls who are interested in mathematical sciences might not be lower now than they were when I grew up. And balancing career and family remains a big challenge. It makes most women face difficult decisions which usually compromise their work," she said.

Frances Kirwan at Oxford University, one of Britain's leading mathematicians, said: "Maths is a hugely rewarding subject, but sadly many children lose confidence very early and never reap those rewards. It has traditionally been regarded as a male preserve, though women are known to have contributed to its development for centuries – more than 16 centuries if we go back to Hypatia of Alexandria.

"In recent years around 40% of UK undergraduates studying maths have been women, but that proportion declines very rapidly when we look at the numbers progressing to PhDs and beyond. I hope that this award will inspire lots more girls and young women, in this country and around the world, to believe in their own abilities and aim to be the Fields medallists of the future."