Questions: Definable Closure and Algebraic Independence

Question 1

An element b satisfies exactly 3 distinct solutions to the formula φ(x, a₁, a₂) with parameters from A. Which closure contains b?

• A. dcl(A) only — b is uniquely specified by the formula
• B. Neither dcl(A) nor acl(A) — b must be the unique solution to be in either
• C. acl(A) but not dcl(A) — b is in a finite definable set but is not uniquely pinned
• D. acl(A) and dcl(A) — any element appearing in a definable formula is in both

Answer: (C)

dcl(A) requires b to be the *unique* solution — if φ has multiple solutions, b is not definitionally determined. acl(A) only requires b to lie in a *finite* definable set, and a set of 3 elements qualifies. So b ∈ acl(A) but b ∉ dcl(A). This is why dcl(A) ⊆ acl(A) always, but the containment can be strict.

Question 2

In which model does acl(A) = A for any parameter set A (i.e., the algebraic closure adds nothing)?

• A. An algebraically closed field — every finite set is already closed
• B. A dense linear order without endpoints — no finite set of points is definable from others
• C. A vector space over Q — linear combinations generate all elements algebraically
• D. The integers ℤ — the division algorithm collapses all ideals to principal ones

Answer: (B)

In a dense linear order (like ℚ with the usual ordering), for any finite set of parameters A, the only elements whose membership in a finite definable set is forced are the parameters themselves. You cannot trap a new point in a finite definable set using only finitely many endpoints in a dense order. By contrast, in an algebraically closed field, acl(A) is the full field-theoretic algebraic closure — much larger than A.

Question 3

In a stable theory, a set B is independent from C over A if and only if no element of B is algebraic over A ∪ (C minus that element).

• T. True
• F. False

Answer: True

This is the correct model-theoretic definition of independence over A. It directly mirrors field-theoretic algebraic independence: a set of elements is algebraically independent over F if no element satisfies a polynomial over F and the remaining elements. In stable theories, forking captures exactly this algebraic constraint — an element forks over A iff it is algebraically constrained by the new parameters beyond what A already forced.

Question 4

The definable closure dcl(A) and the algebraic closure acl(A) usually coincide in any first-order model.

• T. True
• F. False

Answer: False

dcl(A) ⊆ acl(A) always, but they need not be equal. dcl(A) requires the element to be the *unique* solution to some formula with parameters from A. acl(A) only requires the element to lie in a *finite* definable set — it may not be pinned down uniquely. Dense linear orders provide the clearest example: acl(A) = A (no new elements), but in an algebraically closed field, acl(A) properly contains dcl(A) since algebraic elements satisfying polynomials of degree > 1 are in acl but not dcl.

Question 5

Why does the model-theoretic independence relation (via forking) qualify as a genuine dimension theory, and what algebraic notion does it generalize?

Model answer: Model-theoretic independence generalizes algebraic independence from field theory. In a stable theory, forking captures when an element is 'algebraically constrained' by new parameters beyond a base set A. A basis (maximal independent set) can be defined just as in field theory, and all bases have the same cardinality — the dimension of the model over A. This mirrors the vector space theorem that all bases have the same size, and the field-theory result that all transcendence bases of an extension have the same transcendence degree. The machinery works for any stable theory, not just fields.

The point is that dcl, acl, and independence together reconstruct 'linear algebra over a model.' The key steps are: (1) acl generalizes algebraic closure, (2) independence generalizes algebraic independence (no element is acl-dependent on the others), and (3) all maximal independent sets have the same cardinality (like vector space dimension). This gives a dimension notion valid in any stable model — including theories of graphs, groups, and combinatorial structures — wherever forking is well-behaved.