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I don't know what "dominating" means so perhaps this isn't a counterexample, but how about $-p(x)=x^5 - x^4 - x^3 - x^2 + x + 1=(x^2-x-1)(x^3-1)$ (which has a unique real root greater than 1), with $a …

Joel -- it's difficult to work out what you're asking. Of course both possibilities can occur, as Wanax said. Furthermore both possibilities can occur even for the same modular form. For example, if y …

If all the Galois conjugates of an algebraic integer $\alpha$ have absolute value at most 1, then the norm of this algebraic integer is a rational integer with absolute value at most 1. Hence either t …

The odd-order theorem states that every finite group of odd order is solvable, and the proof involves developing a very large theory explaining what the smallest counterexample looks like, and to ulti …

Here is a completely different kind of answer to this question. A perfectoid space is a term of type PerfectoidSpace in the Lean theorem prover. Here's a quote from the source code: structure perfe …

It might all depend on precisely what you mean by Serre's conjecture. Various versions are in print. Serre's original conjecture stayed away from $k=1$ and K-W resolved this version of the conjecture …

Just to add one more thing to what Pete said: the variety A_f that one normally attaches to f might have endomorphism ring bigger than an order in the coefficient field of f: for example if E is an el …

Let $K$ be a non-arch local field (I'm only interested in the char 0 case), let $\mathbb{G}$ be a connected reductive group over $K$ and let $G=\mathbb{G}(K)$. If $V$ is a smooth irreducible complex r …

OK so take the unique tree on 3 vertices. Claim: you can't encode this with an arithmetic progression (AP). For if the AP is $a,a+d,a+2d$ then (because we have two edges) either vertices 1 and 2 are j …

The following slip on p82 was found by Kenny Lau when he was formalising Prop 7.8 in Lean: In the line "Substituting (1) and making repeated use of (2) shows that each element of C is..." there's an i …

I am not so sure of the US system but one of the books we recommend at our university is Martin Liebeck's "A concise introduction to pure mathematics". http://www.amazon.co.uk/Concise-Introduction-Pu …