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Let $\mathbb{Z}^{n}$ be the free abelian group of rank $n$. A ring structure on $\mathbb{Z}^{n}$ is a choice of a unit element $e\in \mathbb{Z}^{n} $ and a bilinear map $m:\mathbb{Z}^{n}\otimes_{\mat …

This question is related to this one. Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from the algebraic closure of the field of rational numbers to the field of complex numbers. Is …

Any finite field extension (in particular Galois extension) of $\mathbb{Q}$ is ramified. Is there an intuitive geometric explanation of this fact? Suppose we have an number field $K$, is any Galois e …

I would like to know what is the recent progress about the group homomorphism $$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$ $\mathrm{Gal}(\overline{\mathbf …

Is the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})$ the same as the group $\mathrm{Aut}_{\mathbf{Q}}(\overline{\mathbf{Q}})$ the automorphism group in the category of $\mathb …

Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from an algebraic closure of the field of rationals to the field of complex numbers. Question 1: Is it true that $\mathbf{C}$ is isom …

Is there a basic arithmetic statement which is known to be undecidable ? By basic arithmetic statement I do mean an easy statement in the spirit of the Collatz conjecture . By the way is there some …

Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves" I hope I'm not distorting his phrase, can someone expla …

I was following some lectures by Edward Frenkel about Langlands correspondence. He was describing some analogies between number theory and theoretical physics (Mirror symmetry). At some point ( my lac …

I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to le …