## Student Blog: Ainsley Pullen

By Ainsley Pullen

If you’ve heard the name Évariste Galois before it’s probably for one of two reasons. The first is his tragic early death at the age of twenty in a duel. The second is his remarkable result that quintics are not solvable by radicals. Quintics, are fifth order polynomials. Smaller polynomials can be solved for a variable by an expression using radicals, i.e. nth roots. The most familiar example of this is the notorious quadratic equation. It is worth noting that this applies to the “general” quintic because there are some cases where x can be computed with radicals such as f(x)=(x-1)(x-2)(x-3)(x-4)(x-5) and even less trivial examples.

This result was also proved independently by Abel but what makes Galois’ proof so interesting is the way in which he solved the problem. He did it by further developing the notion of groups, even coining the term ‘group’. Specifically, he introduced the concept of a normal subgroup, which is a subgroup invariant under conjugation. This means a subgroup H is normal if gH=Hg for any element, g, from the group. Normal subgroups are used in the construction of quotient groups which I worked with during my project. Group Theory is now a fundamental part of undergraduate curriculum with Galois Theory in particular forming an active area of further research. Not bad for a young French boy, who only wrote a few manuscripts –some written just days before he died.

But Galois wasn’t just a mathematician that died young, he had a turbulent political life that likely led him to that fateful duel. His father was mayor and leader of the liberal party of their town who committed suicide due to a slander campaign by the local priest. This occurred not long before Galois failed the university admittance exam due to lack of explanations. Even after he was admitted Galois’ university career was not long as his criticism of the university director got him expelled. He then joined the Republican (i.e. anti-monarchist) guard which was soon after disbanded. Galois was arrested (for the second time) when he lead a protest, heavily armed and in his illegal uniform. He was released just 1 month before his death. His funeral had a strong police presence due to fears it would spark a revolution but this was delayed until the day of Senator Lamarque’s death (as told in Les Misérables). He was buried in an unmarked grave but there is a small memorial on his father’s grave in the town Galois grew up in.

To say that Galois lead me to majoring in mathematics would discredit my family, a little bit of genetics and some great teachers…
To say that Galois was my first mathematical love would discredit prime numbers, suduko and again some great teachers…
But he will always have a special place in my heart.

Ainsley Pullen was one of the recipients of a 2015/16 AMSI Vacation Research Scholarship.