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Abstract Algebra - Part II

Problem 1 (DF, Page 519, Number 7)

Prove that $x^3-nx+2$ is irreducible unless $n=-1,3,5$.

Problem 2 (DF Page 519, Number 2)

Let $\theta$ be a root of $x^3-2x-2$. Write $\frac{1+\theta}{1+\theta+\theta^2}$ in the form $c_0+c_1\theta+c_2\theta^2$.

Problem 3 (DF, Page 529, Number 1)

Let $F$ be a finite field of characteristic $p$. Prove that $F$ has $p^{n}$ elements for some positive integer $n$.

Problem 4 (DF, Page 530, Number 4)

Find the degree over $\Q$ of $2+\sqrt{3}$ and $1+\sqrt[3]{2}+\sqrt[3]{4}$.

Problem 5 (DF, Page 530, Number 7)

Prove that $\Q(\sqrt{2}+\sqrt{3})=\Q(\sqrt{2},\sqrt{3})$. Conclude that $[\Q(\sqrt{2}+\sqrt{3}):\Q]=4$. Find a minimal polynomial over $\Q$ for $\sqrt{2}+\sqrt{3}$.

Problem 6 (DF, Page 530, Number 15)

A field $F$ is called formally real if $-1$ is not expressible as a sum of squares in $F$. Let $F$ be a formally real field, let $f(x)$ be an irreducible polynomial of odd degree, and let $\alpha$ be a root of $F$. Prove that $F(\alpha)$ is also formally real.

Hint: Choose a counterexample $f(x)$ of minimal degree. Show that there is a $g(x)$ of odd degree less than the degree of $f(x)$, and polynomials $p_{1}(x),\ldots, p_{m}(x)$ such that

\[-1+f(x)g(x) = p_{1}(x)^{2} + ... + p_{m}(x)^{2} .\]

Show that $g(x)$ has a root $\beta$ of odd degree over $F$ and $F(\beta)$ is not formally real, contradicting minimality of $f(x)$.

The End

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