You want to adjoin both √2 and √3 to the rationals, forming Q(√2, √3). You know [Q(√2) : Q] = 2 and [Q(√2, √3) : Q(√2)] = 2 (since √3 ∉ Q(√2)). What is [Q(√2, √3) : Q]?
A2 — because you only added two irrational numbers to Q
B3 — because the tower has three fields: Q, Q(√2), and Q(√2, √3)
C4 — by the multiplicative property: [K:E]·[E:F] = 2·2 = 4
D6 — because the degrees of the individual extensions add: 2 + 2 + 2
The tower law (multiplicative property) says [K : F] = [K : E] · [E : F] for a chain F ⊆ E ⊆ K. Here [Q(√2,√3) : Q(√2)] = 2 and [Q(√2) : Q] = 2, so [Q(√2,√3) : Q] = 2 · 2 = 4. Option B confuses the number of fields in the tower with the degree. Option D confuses multiplication with addition — degrees multiply, not add.
Question 2 Multiple Choice
If [K : F] = 7, what can you conclude about intermediate fields E with F ⊊ E ⊊ K?
AThere is exactly one intermediate field, since 7 = 1 + 6
BThere are no intermediate fields — the multiplicative property forces [K:E]·[E:F] = 7, but 7 is prime, so neither factor can be between 1 and 7
CThere are at most 7 intermediate fields, one for each divisor of [K:F]
DWe cannot conclude anything without knowing which fields K and F are
By the tower law, [K:E]·[E:F] = [K:F] = 7. Since 7 is prime, the only factorizations are 1×7 and 7×1. The factor 1 would mean E = F or E = K — not a proper intermediate field. So there is no intermediate field strictly between F and K. This is a powerful consequence: prime degree forces a 'gap' in the lattice of subfields.
Question 3 True / False
If F ⊆ E ⊆ K with [K : E] = 3 and [E : F] = 4, then [K : F] = 7.
TTrue
FFalse
Answer: False
The tower law says [K : F] = [K : E] · [E : F] = 3 · 4 = 12, not 7. Adding the degrees is a common error. The multiplicative property uses multiplication because a basis for K over F is constructed by combining a basis for E/F (4 elements) with a basis for K/E (3 elements), giving 4 × 3 = 12 basis elements in total.
Question 4 True / False
Every element of Q(√2) can be written uniquely as a + b√2 where a, b ∈ Q.
TTrue
FFalse
Answer: True
The set {1, √2} is a basis for Q(√2) as a vector space over Q. This means every element has a unique representation as a linear combination a·1 + b·√2 with rational coefficients a and b. Uniqueness follows from the fact that {1, √2} is linearly independent over Q (if a + b√2 = 0 with a, b ∈ Q and b ≠ 0, then √2 = −a/b would be rational, a contradiction). The degree [Q(√2) : Q] = 2 is precisely the size of this basis.
Question 5 Short Answer
How does the tower law (multiplicative property) allow mathematicians to prove that certain classical geometric constructions — like trisecting an arbitrary angle — are impossible with compass and straightedge?
Think about your answer, then reveal below.
Model answer: Compass-and-straightedge constructions correspond to field extensions of degree 2 (each step adjoins a square root). A constructible length must lie in a field of degree that is a power of 2 over Q. Trisecting a general angle requires solving a cubic equation, which would produce an extension of degree 3. By the tower law, any intermediate field in a tower of quadratic extensions has degree that is a power of 2, so a degree-3 extension cannot appear in such a tower. Therefore trisection is impossible — it would require an extension whose degree (3) does not divide any power of 2.
The tower law makes degree arithmetic rigorous: it tells us exactly which degrees are achievable by chaining extensions. Since 3 is not a power of 2, no sequence of square-root constructions can produce an angle trisection. The same argument applies to doubling the cube (requires ∛2, degree 3) and squaring the circle (requires π, which is transcendental and lies in no finite-degree extension of Q).