Math 262

Problem set 1, due Wednesday, January 21

1. Can you find a field with three elements? With 4 elements? [Extra credit: for which integers n > 1 can you find a field with n elements?]

2. Let R denote the field of real numbers. Let R^{infinity x infinity} denote the set of all ``infinite matrices'' with entries in R; this is the set of all arrays of real numbers of the form:

a11 a12 a13 ...
a21 a22 a23 ...
a31 a32 a33 ...
 .   .   .
 .   .   .
 .   .   .

3. Write down addition and multiplication tables for the ring Z/6Z.

[optional: Suppose that K is a commutative ring. Let K[[x]] denote the set of ``formal power series'' with coefficients in K: these are all things of the form

• a₀ + a₁ x + a₂ x² + ... + a_n xⁿ + ...

4. Do problems 3 and 12 from Section 5.2 in the book.

5. Find some properties of the determinant. You might look in the book at Sections 5.2-5.4, both the text and the problems. Try to figure out how to prove them. By mid-day on Tuesday, let me know (by email, for example) what properties you have found. On Wednesday, don't turn turn them in, but instead be prepared to present your findings to the rest of the class.

Some properties that no one's mentioned yet:

• det(A) = det(B) if B is obtained from A by adding a scalar multiple of one row of A to another.
• Equation (5-19) (on page 157 in my copy of the book).
• det(AB) = det(A) det(B).
• det(A^t) = det(A), where A^t denotes the transpose of A. (Be warned that this one probably requires the use of permutations.)
• Assume the previous two properties and do Exercise 4 in Section 5.4.

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