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Midterm (25th Feb 2020)

Problem 1

1. Give the definition of a group

2. Construct a Caylay table for $(\mathbb{N_7},+)$ and $(\mathbb{N_7, \cdot})$, with addition and multiplication modulo 7.

3. Using the Caylay table for $(\mathbb{N_7}, +)$, verify that it is a group.

4. Using the Caylay table for $(\mathbb{N_7}, \cdot)$, verify that it is a group.

5. Find a normal subgroup $N$ of $(\mathbb{N_7},+)$

6. Construct the factor group $(\mathbb{N_7},+)/N$. Explain how to perform addition in the factor group.

7. Find the index and the order of of $N$ inf $(\mathbb{N_7},+)$

8. Find the generators of $(\mathbb{N_7},+)$ and $(\mathbb{N_7},\cdot)$

Problem 2

Consider the polynomials $f(x) = 2x^5+x + 2$ and $g(x) = x^3+2x+2$ in $\mathbb{F_7[x]}$

1. Give the definition of a monic polynomial. Are $f(x)$ and $g(x)$ monic?

2. Give the definition of the degree $deg(f)$ of polynomial $f(x)$. What at the degrees of $f(x)$ and $g(x)$?

3. Compute the sum $f(x) + g(x)$

4. Compute the product $f(x) \cdot g(x)$

5. Divide $f(x)$ by $g(x)$ with remainder, i.e find polynomials $q(x)$ and $r(x)$ in $\mathbb{F_7[x]}$ s.t $f(x) = q(x) \cdot g(x) + r(x)$ and $deg(r) < deg(g)$

Problem 3

Consider the polynomial ring $\mathbb{F_2[x]}$

1. Give the definition of a reducible polynomial.

2. Find all irreducible polynomials in $\mathbb{F_2[x]}$ of degree 3.

3. Take $p(x)$ to be one of the irreducible polynomials from task 2. Use it to construct the extention field $\mathbb{E} = \mathbb{F_2[x]}/(p(x))$

4. Consider the elements $a = [x + 1], b = [x^2 + x]$ and $c = [x^2 + x + 1]$ of $\mathbb{E}$. Compute the sums: $a + b, a + c, b + c$ and the products $a \cdot b, a \cdot c$ and $b \cdot c$

5. What are the addative inverses of $a, b$ and $c$?

6. What would an element $d \in \mathbb{E}$ have to satisfy in order for it to be a multiplicative inverse to e.g $a$?