Tag Archives: Chris Smyth

Schur-Siegel-Smyth-Serre-Smith

Posted on November 25, 2021 by Persiflage

If \(\alpha\) is an algebraic number, the normlized trace of \(\alpha\) is defined to be \( \displaystyle{T(\alpha):=\frac{\mathrm{Tr}(\alpha)}{[\mathbf{Q}(\alpha):\mathbf{Q}].}}\) If \(\alpha\) is an algebraic integer that is totally positive, then the normalized trace is at least one. This follows from the AM-GM … Continue reading →

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Vesselin Dimitrov on Schinzel–Zassenhaus

Posted on February 10, 2020 by Persiflage

Suppose that \(P(x) \in \mathbf{Z}[x]\) is a monic polynomial. A well-known argument of Kronecker proves that if every complex root of \(P(x)\) has absolute value at most 1, then \(P(x)\) is cyclotomic. It trivially follows that, for a non-cyclotomic polynomial, … Continue reading →

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Abelian Spiders

Posted on January 11, 2015 by Persiflage

This is a blog post about the thesis of my student Zoey Guo, who is graduating this year. (For a blog post on the thesis of my other student graduating this year, see this.) Let \(\Phi\) be a finite graph. … Continue reading →

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The Abelian House is not closed

Posted on August 3, 2014 by Persiflage

Today I will talk about \(\displaystyle{\frac{97 + 26 \sqrt{13}}{27} = 7.064604\ldots}\) For an algebraic integer \(\alpha\), the house \(\overline{|\alpha|}\) is the absolute value of the largest conjugate of \(\alpha.\) Kronecker proved the following: If \(\overline{|\alpha|} \le 1\), then either \(\alpha … Continue reading →

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