Prove the following identity

the sum running through and the matrix being of size .

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#
linear algebra

# [Some punny title about determinants and fixed points]

# Its hip to be skew

# The order of an automorphism

# Keeping it real

# A generic curve

# Polynomial Decomposition

# Sum of powers

Prove the following identity

the sum running through and the matrix being of size .

Show that the determinant of a skew symmetric matrix is never reducible when viewed as a polynomial in variables.

Let . What is the maximum order can have?

Let be a matrix with a real, simple eigenvalue. Prove that there exists a neighborhood of in which all matrices have at least one real eigenvalue.

Exhibit a curve in such that any -uple of points in its trace is in general position.

Decide if there are polynomials over such that

.

Provide the most elegant proof to the following claim you can produce. Suppose and are nonzero complex numbers and for every positive integer it is true that . then and the are a reordering of the .