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  • What does ramification have to do with separability?
    A finite map f:X -> Y is totally ramified at y if the scheme theoretic fibre X_y -> y is a universal homeomorphism If k is a field, and A is a finite k-algebra, then A is totally ramified over k in the above sense if and only if a) A is local, and b) the last condition holds after all base changes on k
  • ramification - Ramified quaternion algebras - MathOverflow
    $\begingroup$ I neglected to mention that later in the book I'm reading Maclachlan Ried actually do devote a whole chapter to orders in quaternion algebras, they just give an overview of it early on
  • Can every genus $2$ curve be written as ramified cover of elliptic curve?
    Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers
  • A type of principal ideal theorem of class field theory for ramified . . .
    For ramified primes, if we take the ground field to be the rationals, when in a cyclotomic field at leat two primes ramified, then the prime ideals over them are not in general principal (but their product is, see my second question) Also for my first question, totally ramifiedness is not solved for me Thanks $\endgroup$ –
  • What are the primes that are ramified? - MathOverflow
    $\begingroup$ Probably one can show that if $\mathfrak p^v$ divides $\mathfrak c$ and $\left|\left( \mathcal O_K \mathfrak p^v\right)^\times \right| > | \mathcal O_K^\times |$ then $\mathfrak p$ is in fact ramified, which combined with what TKe says handles everything except for very small primes, which can be done explicitly $\endgroup$
  • Definition and sigularity of Ramified covers - MathOverflow
    By the book of Kollár and Kovács (See Page 65-65), it claims that the discrepancy does not get worse by taking a finite ramified cover (in their definition) I looked at the proof, and feel it could go through without any change for the (general ) cyclic ramified cover case Did I miss something?
  • Branch loci of Ramified covers - MathOverflow
    Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers
  • Explicit description of a quaternion algebra with a prescribed set of . . .
    It's much easier if the requirements are not exact For simplicity, I'm going to find a quaternion algebra defined over $\mathbb Q$ and tensor up to $\mathbb Q(\sqrt[4]{2})$ Take a split prime, say $73$ Then a quaternion algebra ramified at $73$ will remain ramified in the extension
  • Are all totally ramified $\\mathbb{Z}_p$-extensions of local fields . . .
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  • Infinite tamely ramified $p$-extensions of $\\mathbb{Q}$ contain . . .
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