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Galois died in a duel at the age of twenty. Yet, he gave us what we now call Galois theory. It decides all three ancient classical problems, squaring the circle, doubling the cube, and partitioning angles into three equal parts, all with compass and ruler alone. Galois theory also tells us that there is no general formula to solve the integer equation

$$

x^5+a_1x^4+a_2x^3+a_3x^2+a_4x+a_5=0

$$

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$$

x^5+a_1x^4+a_2x^3+a_3x^2+a_4x+a_5=0

$$

Continue reading...

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