Prove that is surjective for every integer but is not necessarily surjective.

# linear algebra

# An exercise in Linear Algebra

Let be a field of characteristic and be the -vector space of homogeneous degree polynomials in variables. Show that the linear span of the set of -th powers of linear polynomials is the whole space of homogeneous degree polynomials.

In coordinate-free terms: if is a finite-dimensional -vector space then spans .

# [Some punny title about determinants and fixed points]

Prove the following identity

the sum running through and the matrix being of size .

# Its hip to be skew

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

# The order of an automorphism

Let . What is the maximum order can have?

# Keeping it real

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.

# A generic curve

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