# HW Number 2

## Due on March 11th

1. Find a quadratic polynomial $$f \in \mathbb{F}_7[x]$$ such that $$f \equiv x^5 + 3x^4 +5x + 1 \mod{(x^3 + x + 1)}$$
2. Let $$G$$ be the group formed by all rigid motion symmetries of an equilateral triangle (imagine a solid metal triangle with corners labelled A,B,C for instance).
• Describe the elements of $$G$$.
• Is this group commutative, that is $$a * b = b * a$$ for all $$a,b \in G$$?
• Is this group somehow equivalent to the group of all permutations of three items?
• Does this group have a generator?
3. Let $$H = \{ (0, 0, 0, 0, 0, 0), (1, 1, 1, 0, 0, 0), (0, 0, 0, 1, 1, 1), (1, 1, 1, 1, 1, 1)\}$$ be a subgroup of $$\mathbb{F}_2^6$$ under vector addition modulo 2. Find all of the cosets of $$H$$. Find the element of each coset with smallest Hamming weight.
4. Find a basis of the nullspace of $$\left(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 \end{array}\right)$$ over $$\mathbb{F}_2$$.