توصيفگر ها :
تعريف‌پذيري , ميدان ارزيابي هنسلي , ميدان‌هاي شبه‌بسته‌ي جبري , ميدان‌هاي متناهي
چكيده فارسي :
چكيده:
هدف اصلي اين پايان نامه اثبات قضيه ي زير است:
«فرض كنيد K يك ميدان ارزيابي هنسلي با حلقه ي ارزياب O و ميدان باقيمانده هاي F باشد. اگر ميدان
F متناهي يا شبه بسته ي جبري باشد، آنگاه حلقه ي ارزياب O در ميدان ارزيابي هنسلي K به صورت وجودي
و بدون پارامتر در زبان حلقه ها تعريف پذير است».
براي اثبات اين قضيه ابتدا دو حكم كلي زير را اثبات مي كنيم:
.1 اگر زيرمجموعه ي U از حلقه ي ارزياب O شامل ايده آل ماكزيمال m باشد و O ⊆ T همه ي كلاس هاي
باقيمانده را قطع كند، تساوي T + U = O برقرار است.
.2 اگر چندجمله اي [X[O) ∈ X(f ويژگي هاي مطلوبي داشته باشد، زيرمجموعه ي تعريف پذير
} = Uf از حلقه ي ارزياب O شامل ايده آل ماكزيمال m است.
1
f(x) −
1
f(y)
|x, y ∈ K}
سپس براي دو حالت ميدان باقيمانده هاي متناهي و ميدان باقيمانده هاي شبه بسته ي جبري، به صورت متفاوت
وجود چندجمله اي f را اثبات مي كنيم. در نهايت نشان مي دهيم كه در حالت ميدان باقيمانده هاي متناهي و
T = {x ∈ K : x
ميدان باقيمانده هاي شبه بسته ي جبري به ترتيب زيرمجموعه ها ي تعريف پذير {0 = x − q
(K(f = Tf از حلقه ي ارزياب O، همه ي كلاس هاي باقيمانده را قطع مي كنند.
−1
f(K)
و {0} ∪ −1
قضيه ي ياد شده در مقاله ي [7] با عنوان دقيق زير اثبات شده است.
Fehm, Arno. Existential ∅-definability of henselian valuation rings. The Journal of
Symbolic Logic, 80(1):301–307, 2015
رده بندي موضوعي: 60 C 03
واژگان كليدي: تعريف پذيري، ميدان ارزيابي هنسلي، ميدان هاي شبه بسته ي جبري، ميدان هاي متناهي
چكيده انگليسي :
Abstract:
This M.Sc. thesis is based on the following paper:
• Fehm, Arno. Existential-definability of henselian valuation rings. The Journal of Symbolic Logic, 80(1):301–307, 2015.
Suppose that M is a first-order structure with domain M, A is a given subset of M, and
n is a natural number. A set X ⊆ Mn
is called definable in M with parameters from
A if there exists a formula φ(x1, ..., xn, y1, ..., ym) and elements b1, . . . , bn ∈ A such that
X = {(a1, ..., an) ∈ Mn
| M |= φ(a1, ..., an, b1, ..., bm)}. Identifying definable sets, and the
complexity of their definition, in a particular first order structure is of crucial importance in
model theory. This thesis concerns the definability of valuation ring in a henselian valued field,
where the residue field is finite or pseudo-algebraically closed.
A field K is pseudo-algebraically closed if for every absolutely irreducible polynomial f ∈ K[X, Y ]
there is a point (a, b) ∈ K2 with f(a, b) = 0.
Assume that K is a field and Γ is an ordered abelian group. A valuation map v : K → Γ ∪ {∞}
is a map that satisfies the following properties for all x, y in K:
1. v(x + y) ≥ min{v(x), v(y)}
2. v(x · y) = v(x) + v(y)
3. x = 0 ⇔ v(x) = ∞.
The set O = {x ∈ K : v(x) ≥ 0} is a called a valuation ring of K, and the pair (K, O) is called a
valued field. The ring O is local, that is it has a unique maximal ideal m = {x ∈ K : v(x) > 0}.
The field F =
O
m
is referred to as the residue field. The canonical image of an element a ∈ O in
F is denoted by a¯.
A valued field (K, O) is called henselian if for each f ∈ O[X] and a ∈ O with ¯f(¯a) = 0 and
¯f
′
(¯α) ̸= 0 in the residue field, there exists some b ∈ O such that f(b) = 0 and ¯b = ¯a.
The main theorem of the thesis is the following:
Let K be a henselian valued field with valuation ring O and residue field F. If F is
finite or pseudo-algebraically closed and the algebraic part of F is not algebraically
closed, then there exists an existential definition of O in K, with no parameters.
To establish this theorem, we first verify that:
1. If U, T ⊆ O are such that m ⊆ U and T meets all residue classes (i.e. T¯ = F ), then
O = U + T.
2. If f ∈ O[X] is a monic polynomial such that ¯f has no zero in F, and a ∈ O is such that
f
′
(a) ∈/ m, then U := f(K)
−1 − f(K)
−1
satisfies m ⊆ U ⊆ O.
We deduce the existence of the polynomial f in different ways for cases of finite and pseudoalgebraically closed residue field. Finally, we show that when F is finite (pseudo-algebraically
closed) the definable subset T = {x ∈ K : x
q − x = 0} (Tf = f(K)
−1f(K)
−1 ∪ {0}) of Off meets
all residue classes.
