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comment Is an extension of a discrete absolute value discrete too?
You could take a look at Lang's Algebra, Chapter XII. However the theory there is written in the language of valuations, that is instead of a non-archimedian absolute value $v$ the map $-\log\circ v$ is considered. Another source is the book Valuation Theory by O. Endler, which sadly is out of print now.
Oct
23
answered Is an extension of a discrete absolute value discrete too?
Oct
13
awarded  Yearling
Oct
5
comment It's true that a valuation ring $R$ in the quotient field of a normal ring $A$ contain $A$?
The definition Jyrki gives for $v_\infty$ requires $p\neq 0$. For $p=0$ the map $v_\infty$ is defined to be equal to $\infty$. Nothing to check here. Moreover: the valuation ring of $v_\infty$ does not contain $k[x]$ but $k[x^{-1}]$. The latter is a polynomial ring too, and $v_\infty$ ist just the $x^{-1}$-adic valuation from that point of view.
Oct
5
comment Ring homomorphism between real numbers and real valued functions
In my eyes the substitution principle is mentioned to make clear, that a formal polynomial can be considered as a map $\mathbb{R}^n\rightarrow\mathbb{R}$, which is a priori not clear considering the algebraic construction of polynomials.
Oct
4
answered Ring homomorphism between real numbers and real valued functions
Oct
4
answered Birational map and birational morphism in algebraic geometry
Oct
1
comment greatest common divisor in the ring $F(x)[y]$
It was guessing - the elements are quite simple. There is however a method to find the factors: the (extended) Euklidean algorithm, which essentially consists of a sequence of polynomial divisions.
Sep
30
revised greatest common divisor in the ring $F(x)[y]$
added 87 characters in body
Sep
30
answered greatest common divisor in the ring $F(x)[y]$
Sep
28
answered Completion of absolute value on an integral domain
Sep
28
answered Unique factorization into irreducible elements?
Sep
21
awarded  Enlightened
Sep
21
awarded  Nice Answer
Sep
20
answered Generator of a subalgebra over a ring
Sep
18
revised local dimension of irreducible varieties
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Sep
18
answered local dimension of irreducible varieties
Sep
15
answered The meaning of this definition?
Aug
21
answered Generators of the Relations of a Galois Extension
Aug
6
answered Contrasting definitions of bimodules? An illusion?