Algebra
What this domain tests
Algebra on PEI Math 521B centers on structure: functions, polynomials, and the algebraic properties of operations. Questions typically ask you to verify whether a set with an operation forms a group or ring, analyze polynomial roots and factorization, work with functional equations, or justify inequalities. Precision about quantifiers and domains is what separates full marks from partial credit.
Core rules and concepts
- Group axioms: closure, associativity, identity, and inverses. A subset H of a group G is a subgroup exactly when it is nonempty and closed under the operation and inverses.
- Rings and fields: a field requires every nonzero element to have a multiplicative inverse. The integers mod n form a field precisely when n is prime.
- Factor theorem and remainder theorem: for a polynomial p over a field, p(a) = 0 iff (x − a) divides p(x); the remainder on division by (x − a) is p(a).
- Rational root theorem: any rational root p/q in lowest terms of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient.
- Functional equations: substituting special values (0, 1, −x) and comparing instances is the standard technique; injectivity or surjectivity claims need proof, not assumption.
Worked example
Find all rational roots of f(x) = 2x³ − 3x² − 8x + 3. Candidates are ±1, ±3, ±1/2, ±3/2. Test x = 1: 2 − 3 − 8 + 3 = −6, not a root. Test x = 3: 54 − 27 − 24 + 3 = 6, not a root. Test x = 1/2: 2(1/8) − 3(1/4) − 4 + 3 = 0.25 − 0.75 − 1 = −1.5, not a root. Test x = −1/2: 2(−1/8) − 3(1/4) + 4 + 3 = −0.25 − 0.75 + 7 = 6, not a root. Test x = 3/2: 2(27/8) − 3(9/4) − 12 + 3 = 6.75 − 6.75 − 9 = −9. Test x = −3/2: 2(−27/8) − 3(9/4) + 12 + 3 = −6.75 − 6.75 + 15 = 1.5. Test x = −3: −54 − 27 + 24 + 3 = −54. None of the candidates vanish, so f has no rational roots; over the rationals it is irreducible into a cubic. The lesson: the theorem narrows the search, it does not guarantee a hit.
Traps that cost marks
- Assuming a functional equation's solution is linear or that f is injective without deriving it.
- Checking closure but omitting inverse or identity verification when testing a subgroup.
- Claiming Z/nZ is a field for composite n, or forgetting that n must be prime.
- Losing sign errors when evaluating candidates like −3/2 in the rational root test.