Algorithmic Number Theory in Function Fields

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References, Henning Stichtenoth Algebraic Function Fields and Codes second. ed GTM vol 54 Springer 2009, Michael Rosen Number Theory in Function Fields GTM vol 210. Springer 2002, Gabriel Daniel Villa Salvador Topics in the Theory of Algebraic. Function Fields Birkha user 2006, Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 2 92. Valuation Theory,Absolute Values,Throughout let F be a field.
Definition, An absolute value on F is a map F R such that for all a b F. a 0 with equality if and only if a 0,a b a b archimedian or. a b max a b non archimedian, The well known absolute value on Q or on R or on C is an. archimedian absolute value in the sense of the above definition. The trivial absolute value on any field F defined via a 0 when. a 0 and a 1 otherwise is a non archimedian absolute value. Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 4 92. p Adic Absolute Values on Q, Let p be ay prime number and define a map p on Q as follows. For r Q write r p n with n Z and p ab and set, Then p is a non archimedian absolute value on Q called the p adic.
absolute value on Q,Theorem Ostrowski, The p adic absolute values along with the trivial and the ordinary absolute. value are the only valuations on Q, Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 5 92. Rational Function Fields,For any field K, K x denotes the ring of polynomials in x with coefficients in K. K x denotes the field of rational functions in x with coefficients in K. K x f x g x K x with g x 0, Note that F K x is our first example of an algebraic function field. More formally,Definition, A rational function field F K is a field F of the form F K x where.
x F is transcendental over K, Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 6 92. Absolute Values on K x,Fix a constant c R c 1 and let r x K x be nonzero. p adic absolute values on K x, Let p x be any monic irreducible polynomial in K x and write. r x p x n a x b x with n Z and p x a x b x Define,r x p x c n. Then p x is a non archimedian absolute value on K x. Infinite absolute value on K x,Write r x f x g x and define.
r x c deg f deg g,Then is a non archimedian absolute value on K x. Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 7 92. Remarks on Absolute Values on K x, These plus the trivial absolute value are essentially all the absolute. values on K x up to trivial modifications such as,using a different constant c. using a different normalization on the irreducible polynomials. All absolute values on K x are non archimedian different from Q. When K Fq is a finite field of order q one usually chooses c q. When K is a field of characteristic 0 one usually chooses. c e 2 71828, Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 8 92. Valuations,Definition, A valuation on F is a map v F R such that for all a b F.
v a if and only if a 0,v ab v a v b,v a b min v a v b. The pair F v is called a valued field,Here n and n for all n Z. Let c 1 be any constant Then v is a valuation on F if and only if. c v is a non archimedian absolute value on F with c 0. Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 9 92. Trivial valuation for any a F define v a when a 0 and. v a 0 otherwise Then v is a valuation on F, p adic valuations on Q for any prime p and r p n a b Q define. vp r n Then vp is a valuation on Q, p adic valuations on K x for any monic irreducible polynomial. p x K x and r x p x n a x b x K x define,vp x r x n Then vp x is a valuation on K x.
Infinite valuation on K x for r x f x g x K x define. v r x deg g deg f Then v is a valuation on K x, Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 10 92. More on Valuations,Definition, A valuation v is discrete if it takes on values in Z and normalized if. there exists an element u F with v u 1 Such an element u is a. uniformizer or prime element for v, All four valuations from the previous slide are discrete. Every p adic valuation on Q is normalized with uniformizer p. Every p adic valuation on K x is normalized with uniformizer p x. The infinite valuation on K x is normalized with uniformizer 1 x. The p adic and infinite valuations on K x all satisfy v a 0 for all. a K They constitute all the valuations on K x with that property. A discrete valuation is normalized if and only if it is surjective. Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 11 92. Valuation Rings, For a discretely valued field F v define the following subsets of F. Ov a F v a 0,Ov a F v a 0,Pv a F v a 0 Ov Ov,Properties.
Ov is an integral domain and a discrete valuation ring i e Ov F. and for a F we have a Ov or a 1 Ov,Ov is the unit group of Ov. Pv is the unique maximal ideal of Ov in particular Fv is a field called. the residue field of v, Every a F has a unique representation a u n with Ov and. Ov is principal ideal domain whose ideals are generated by the. non negative powers of u in particular u is a generator of Pv. Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 12 92. Example p Adic Valuations,For any p adic valuation vp on Q. Ovp r Q r a b with gcd a b 1 and p b,Ov p r Q r a b with gcd a b 1 and p ab. Pvp r Q r a b with gcd a b 1 p a p b,Similarly for any p adic valuation vp x on K x.
Ovp x r x K x r x a x b x with gcd a b 1 p x b x,Ov p x r x K x r x a x b x with gcd a b 1. p x a x b x,Pvp x r x K x x a x b x with gcd a b 1. p x a x p x b x, Fvp x K x p x where p x is the K x ideal generated by p x. Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 13 92. Example Infinite Valuation on K x,For the infinite valuation v on K x. Ov r x K x r x f x g x with deg f deg g,Ov r x K x r x f x g x with deg f deg g.
Pv r x K x x f x g x with deg f deg g,We will henceforth write O P F for brevity. x 7 x 7 1 2 x 3 7x 2,2x 3 3x 2x 3 3x x 2x 3 3, Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 14 92. Definition, A place of F is the unique maximal ideal of a discrete valuation ring in F. The set of places of F is denoted P F, There is a one to one correspondence between the set of normalized. discrete valuations on F and the set P F of places of F as follows. If v is a normalized discrete valuation on F then Pv P F is the. unique maximal ideal in the discrete valuation ring Ov. If P is a place of F then the discrete valuation ring O F containing. P as its unique maximal ideal is determined and P defines a discrete. normalized valuation on F as follows if u is any generator of P then. every element a F has a unique representation a u n with n Z. and a unit in O and we define v a n and v 0 Note that u. is a uniformizer for v, Renate Scheidler Calgary Number Theory in Function Fields UNGC Summer 2016 15 92.
Algorithmic Number Theory in Function Fields Renate Scheidler UNCG Summer School in Computational Number Theory 2016 Function Fields May 30 June 3 2016 References Henning Stichtenoth Algebraic Function Fields and Codes second ed GTM vol 54 Springer 2009 Michael Rosen Number Theory in Function Fields GTM vol 210 Springer 2002 Gabriel Daniel Villa Salvador Topics in the Theory of

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