Questions: Integral Extensions

Question 1

Which of the following elements is integral over ℤ?

• A. 1/2, because it satisfies 2x - 1 = 0
• B. √2, because it satisfies x² - 2 = 0
• C. π, because it satisfies x - π = 0 in ℝ
• D. Both 1/2 and √2 are integral over ℤ

Answer: (B)

An element is integral over ℤ if it satisfies a monic polynomial (leading coefficient 1) with coefficients in ℤ. √2 satisfies x² - 2 = 0, which is monic with integer coefficients. The element 1/2 satisfies 2x - 1 = 0, but this polynomial is not monic (leading coefficient is 2, not 1); no monic integer polynomial has 1/2 as a root. The element π is transcendental over ℚ, so it satisfies no polynomial with rational (let alone integer) coefficients. The distinction between 'root of a polynomial' and 'root of a monic polynomial' is exactly the difference between being algebraic and being integral.

Question 2

If R ⊆ S is an integral extension of integral domains, and 𝔭 is a prime ideal of R, then there exists a prime ideal 𝔔 of S lying over 𝔭 (meaning 𝔔 ∩ R = 𝔭). This is called:

• A. The going-up theorem
• B. The lying-over theorem
• C. The incomparability theorem
• D. Nakayama's lemma

Answer: (B)

The lying-over theorem guarantees that for an integral extension R ⊆ S, every prime ideal of R is the contraction of some prime ideal of S. This is the first of three theorems (lying-over, going-up, incomparability) that describe the relationship between Spec(S) and Spec(R) for integral extensions. It ensures the map Spec(S) → Spec(R) is surjective, which is essential for dimension theory and algebraic geometry.

Question 3

The ring of algebraic integers in ℚ(√-5) is integrally closed.

• T. True
• F. False

Answer: True

The ring of algebraic integers in a number field K is defined as the integral closure of ℤ in K — it consists of all elements of K that are integral over ℤ. By a fundamental theorem, the integral closure of an integrally closed domain in a field extension is itself integrally closed. Since ℤ is integrally closed (its integral closure in ℚ is ℤ itself), the ring of algebraic integers ℤ[√-5] (which equals ℤ + ℤ√-5 in this case) is integrally closed in ℚ(√-5).

Question 4

If b is integral over R and c is integral over R[b], then c is integral over R.

• T. True
• F. False

Answer: True

This is the transitivity of integral dependence. If b satisfies a monic polynomial of degree m over R and c satisfies a monic polynomial of degree n over R[b], then R[b] is a finitely generated R-module (generated by 1, b, ..., b^(m-1)) and R[b, c] is a finitely generated R[b]-module. Therefore R[b, c] is a finitely generated R-module, and since c ∈ R[b, c], c is integral over R by the module-finiteness characterization. This transitivity is crucial: it ensures that the integral closure of R in S is a subring.

Question 5

Explain what it means for a domain R to be integrally closed and give the canonical example of a domain that is not integrally closed.

Model answer: A domain R is integrally closed if every element of its fraction field that is integral over R already belongs to R. Equivalently, the integral closure of R in Frac(R) is R itself. The canonical non-example is ℤ[√-3]: the element (1 + √-3)/2 lies in the fraction field ℚ(√-3) and satisfies x² - x + 1 = 0 (monic, integer coefficients), so it is integral over ℤ[√-3], but it does not belong to ℤ[√-3]. The integral closure is ℤ[(1+√-3)/2], the ring of Eisenstein integers.

Integrally closed domains are the 'non-singular' rings in a precise sense. In algebraic geometry, the coordinate ring of a variety is integrally closed if and only if the variety is normal (no self-intersections or cusps). In number theory, replacing ℤ[√-3] with its integral closure ℤ[(1+√-3)/2] restores unique factorization of ideals (Dedekind domain structure), which is why algebraic number theorists always work with the full ring of integers.