For each part, show that the real number $\alpha$ defined in that part is algebraic over $\mathbb{Q}$.
$\alpha = \sqrt{2 + \sqrt{3 + \sqrt{5}}}$.
$\alpha = \sqrt{2} + \sqrt{3}$.
Let $F$ be a field, let $n$ be a positive integer, and let $p(x) \in F[x]$ be an irreducible polynomial of degree $n$. Define $E = F[x]/\langle p(x) \rangle$, which is an extension field of $F$.
Define the element $\alpha \in E$ by $\alpha = x + \langle p(x) \rangle$ and recall that every element of $E$ can be written as a polynomial in $\alpha$by the proof of Kronecker's theorem; for example, $x^2 + 1 + \langle p(x) \rangle = \alpha^2 + 1$.
Prove that for every element $\beta \in E$ there is one and only one polynomial $r(x) \in F[x]$ such that $\beta = r(\alpha)$ and $\deg r(x) \lt n$.
Hint: $\beta = f(x) + \langle p(x) \rangle$ for some $f(x) \in F[x]$. Apply the polynomial division algorithm.
Use part a to calculate the order (number of elements) of $E$ in the case $F = \mathbb{Z}_p$ where $p$ is a prime number. Justify your answer.
Define $p(x) = x^4 + x + 1 \in \mathbb{Z}_2[x]$ and define the factor ring $E = \mathbb{Z}_2[x]/\langle p(x) \rangle$. Define the element $\alpha = x + \langle p(x) \rangle \in E$.
Show that $p(x)$ is irreducible in $\mathbb{Z}_2[x]$, so that $E$ is a field.
Note: because $\deg p(x) > 3$, it is not enough to show that $p(x)$ has no zeroes in $\mathbb{Z}_2$.
Use problem 2 to list all the elements of $E$ as polynomials in $\alpha$ of degree less than $4$.
Show that $\alpha$ is a generator of the multiplicative group $(E^*,\cdot)$.
Hint: if you show that $\alpha^4$ can be expressed in terms of lower powers of $\alpha$, then you can use this to simplify your computation at every step. Also, it is possible (but not necessary) to save some work by using Lagrange's theorem.
Note: the property of $p(x)$ that you showed in problem 3c is called primitivity. Primitive polynomials are useful in cryptography.
Define $\beta = 2^{1/3} + 2^{2/3}$. Show that $\beta$ is algebraic over $\mathbb{Q}$ and find $\operatorname{irr}(\beta,\mathbb{Q})$ and $\operatorname{deg}(\beta,\mathbb{Q})$.
Hint: write $\beta$ as $\alpha + \alpha^2$ where $\alpha = 2^{1/3}$. Calculate a few powers of $\beta$, using the fact that $\alpha^3 = 2$ to simplify at each step. Show that these powers of $\beta$ are linearly dependent over $\mathbb{Q}$. This will give you a polynomial in $\mathbb{Q}[x]$ that has $\beta$ as a zero. To prove that your polynomial is irreducible, use problem 4a from homework set 6, which is known as the rational root theorem.
Let $F$ be a field, let $E$ be an extension field of $F$, let $\alpha \in E$, and define $F[\alpha] = \{f(\alpha) : f(x) \in F[x]\}$. Assume that $\alpha$ is algebraic over $F$. Use the fact that every nontrivial prime ideal of a principal ideal domain is maximal (see problem 4 from homework set 8) to prove that $F[\alpha]$ is a subfield of $E$.
Note that we already proved this in class; you are asked to give a different proof.
There are only five problems this week.