## Syllabus for the Algebra Qualifying Exam

The algebra qualifying exam covers the following five topics that roughly correspond to five Yale undergraduate or cross-listed Algebra classes. You can use references below as well as the references for the relevant classes (available via Canvas) to prepare.

### 1) Linear algebra, MATH 240.

Linear operators and bilinear forms, tensors. Specific topics include the structure theory

and classification theorems, such as the Jordan normal form theorem, exponentials of linear operators, tensor products, symmetric and exterior powers of vector spaces. References include [L, Ch. 13-16, 19] and [V, Ch. 5,6,8].

### 2) Group theory, MATH 350.

The structure theory and classification of finite groups including p-groups, Sylow theorems, solvable groups. References include [L, Ch. 1] and [V, Ch. 2, S. 10.1-10.5].

### 3) Representations of finite groups, MATH 353/533.

This includes, in particular, basic results (complete reducibility, Schur lemma), group algebra, the classification of finite dimensional irreducible representations, characters, tensor products. References include [L, Ch. 18 and partly Ch. 17], [V, S. 11.1-11.4].

### 4) Fields and Galois theory, MATH 370.

Field extensions, finite fields incl. Frobenius automorphism, Galois groups and the main theorem of Galois theory. Applications incl. solvability by radicals. References are, for example, [L, Ch. 5,6], [V, S. 9.3,10.6,10.7].

### 5) Commutative algebra, MATH 380/500.

Basic constructions with rings, ideals and modules. Principal ideal domains and their modules. Noetherian and Artinian rings and modules. Integral extensions and localization. References include [L, Ch. 2,3,7,10], [V, Ch. 9].

### References:

[L] S. Lang, Algebra, revised 3rd edition, GTM 211, Springer.

[V] E. Vinberg, A course in Algebra. GSM 56, Springer.

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