Maths Project Repository Database

Unique factorisation into primes for the ordinary integers has led some great mathematicians in the past to believe, naively, in unique factorisation for other algebraic integers. One need only note that $(1+\sqrt{5})(1-\sqrt{5})=2×3$ to see that they were wrong. In 1850 Ernst Kummer not only made the mathematical community aware of the failure of unique factorisation, he found a way to measure how much it failed by, and introduced ideal numbers into which algebraic integers do factorise uniquely. In this way he was able to correct some cases of the erroneous proofs that had been predicated on this false assumption.
Possible directions include:

• Give examples of non-unique factorisation.
• Show how ideal numbers arise naturally in this context.
• Describe the arithmetic of ideal numbers and unique factorisation into them.
• Explain the background of why people in the 19th century were trying to use unique factorisation to solve Fermat's last theorem.
• Compare non-unique factorisation to the twist in the Mobius strip.

For MSci students possible directions also include:

• Discuss how this leads into algebraic k-theory.
• Explain how this relates to fibre bundles and topological k-theory.
• Explain the notion of a scheme, where the analogue of an ideal number is a closed subset of a space.
• Give some exposition of Kummer's proof of Fermat's Last Theorem for regular primes