## Evariste Galois (1811 - 1882)

Born in France to a well-educated family, Evariste Galois grew up amid political turmoil.

He took up mathematics in 1823, and his teachers soon recognised his aptitude for the subject.

He was still at school when he began to undertake his own mathematical research, and was only a teenager when in 1829 he published his first paper - on continued fractions - and swiftly followed this with others.

Revolution took France by storm in 1830 and the King fled. Galois was expelled from school for expressing his Republican views in a public letter in the Gazette des Écoles.

He then joined the Artillery of the National Guard, which was later disbanded when the monarchy were reinstated, as they were a threat. Galois was arrested in 1831 for apparent threats against the King, but was acquitted, only to be arrested and imprisoned again on Bastille Day (the 14th July) for wearing the uniform of the Artillery of the National Guard - he was also heavily armed!

In 1832 cholera struck Paris, and Galois, as well as the other prisoners, was transferred to a French boarding house (called a ‘pension’). He was released that same year only to die after being wounded in a duel over Stephanie, the daughter of the physician at the pension. He was such a prominent Republican that his funeral became a republican rally and there was rioting lasting several days.

After his death, his brother and a friend copied out his mathematical papers and sent them to Gauss, Jacobi and others in accordance with Galois’ wishes. The theory outlined in these papers is now called Galois theory and it concerns the groups that arise from the permutations of the roots of equations and other algebraic operations.

## Galois’ mathematics

A continued fraction is a way of writing a number in terms of fractions. It leads to some important mathematics and to some interesting approximations. Here are two (infinite) continued fractions for two important mathematical constants:

\pi\approx3.141592\ldots

\ \ \displaystyle \approx 3+\frac {1}{6+\frac {9}{6+\frac {25}{6+\frac {49}{{6+\frac {81}{\ldots}}}}}}

and

e\approx2.71828\ldots

\ \, \displaystyle \approx 2+\frac {1}{1+\frac {1}{2+\frac {1}{1+\frac {1}{{1+\frac {1}{{4+\frac {1}{\ldots}}}}}}}}

Notice the patterns in the continued fractions and their absence in the decimal expansion. This is frequently true as the continued fraction for the square root of 2 also demonstrates,

\sqrt{2}\approx1.414\ldots

\quad \ \displaystyle \approx 1+\frac {1}{2+\frac {1}{2+\frac {1}{2+\frac {1}{{2+\frac {1}{\ldots}}}}}}

Continued fractions are used to obtain approximations for such numbers. For example, consider the continued fraction,

\displaystyle 1+\frac{1}{1+\frac{1}{1+\frac{1}{\ldots}}}

If we stop the fraction and then evaluate it - inside out as it were - we have an approximation:

{\displaystyle 1+\frac{1}{1+\frac{1}{1+\frac{1}{1}}}}=1+\frac{1}{1+{1\over{2}}}=1+\frac{1}{3\over{2}}=1+{2\over{3}}={5\over{3}}

You might like to try the next approximation

\displaystyle 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1}}}}

and then the next to see what numbers you get. You should find that the numerator and the denominators of these are successive Fibonacci numbers.

Galois’ work was influenced by the developments made by Abel and Jacobi on the theory of equations, the theory of elliptic functions and abelian integrals. But his major contribution was in a mathematical object called a group.

He made important contributions to this area by recognizing that algebraic symmetries ( which sees \sqrt{2} say, as a ‘reflection’ of its algebraic partner -\sqrt{2}), like geometric symmetries and permutations were all examples of the same basic thing.

The structure of the resulting group gives important information about the thing itself.