Standard Generals Questions
Compiled by Kiran S. Kedlaya
The general examination system of the Princeton math department
is unusual in that each student is examined not only in two freely
chosen subjects, but also in three fixed topics: real analysis,
complex analysis, and algebra. The result is a large amount of
repetition of questions in those areas; this page lists a few of
the most popular questions. For more detailed descriptions, please
return to the student guide to generals.
The questions are broken up by topic. Note that "prove" here
usually either means "state without proof" or "sketch a proof
without details".
Algebra
- Prove the Chinese Remainder Theorem.
- What's the Galois group of the polynomial $X^5 - 2$ over $Q$?
State the fundamental theorem of Galois theory.
- Prove the structure theorem for finitely generated abelian
groups (or modules over a principal ideal domain).
- State Schur's Lemma. Prove Maschke's theorem.
- Give a unique factorization domain which is not a principal
ideal domain. Give a ring which is not a unique factorization
domain.
- What does it mean for a matrix to be in Jordan/rational
canonical form?
- Classify all finite fields. Why is the multiplicative group of
a finite field cyclic?
- If $S$ is a skew-symmetric matrix, show that $(I+S)(I-S)^{-1}$
is orthogonal.
Real Analysis
- What is a measurable/Borel
function/set?
- Prove Dini's Theorem: if a sequence $f_n$ of functions on a
closed interval is pointwise decreasing and pointwise convergent
to 0, show that the convergence is uniform.
- Give a counterexample to Dini's theorem when the sequence is
not pointwise decreasing.
- Prove the monotone/dominated convergence theorem.
- When is $L_p$ contained in $L_q$?
- Prove the fundamental theorem of ordinary differential
equations.
- What is the image of $L_1(R)$ under the Fourier transform?
Also, Prove the Plancherel theorem (an L_2 function has the same
norm as its transform).
- Prove the Baire category theorem.
Complex Analysis
- Prove the Arzela-Ascoli
Theorem.
- Prove the Riemann Mapping Theorem. When are two annuli
conformally equivalent?
- Prove the Schwarz Lemma.
- Prove Liouville's Theorem.
- Prove the Phragmen-Lindelöf theorem.
- Prove the little Picard theorem.
- Prove the argument principle.
- Why is differentiable function automatically analytic?
(Morera's theorem)