Articles of abstract algebra

The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational multiples of $\pi$ for which $\cos(\theta)\in \mathbb{Q}$ are $\theta=\frac{\pi}{2},\frac{\pi}{3}$ which corresponds exactly to $n=4,6$ in $\frac{2\pi}{n}$. Can someone give me an explanation for why $\cos(\theta)$ […]

Is $\mathbb{Z} \left$ euclidean?

Is the ring $\displaystyle A=\mathbb{Z} \left[ \frac{1+i \sqrt{7}}{2} \right]$ euclidean? If $N : z \mapsto z \overline{z}$, then for all $z \in \mathbb{C}$ there exists $a \in A$ such that $N(z-a)<1$, except when $z$ has the form $\displaystyle \left(n+\frac{1}{2} \right)+ \left(m+ \frac{1}{2} \right) \frac{1+i \sqrt{7}}{2}$; in this case, you can only find a large inequality. […]

Fraction field of $F(f)$ isomorphic to $F(X)/(f)$

Assume $F$ is a field and $f$ is an irreducible polynomial in $F[X,Y]$ which involves the variable $Y$. Then, by Gauss’s lemma, $f$ is irreducible also in $F(X)[Y]$ so that $F(X)[Y]/(f)$ is a field (where $(f)$ is the ideal generated by $f$). I’m looking for a simple way to see that the fraction field of […]

Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible

Let $R$ be a PID (Principal Ideal Domain) and $x$ is an element R. Prove that the ideal $\langle x\rangle$ is maximal if and only if $x$ is irreducible. Ok, so I know what an irreducible is. I’m thinking that this problem is asking us to set up a proof by contradiction but I can’t […]

Basis for $\mathbb{R}$ over $\mathbb{Q}$

Give me some examples of basis for $\mathbb{R}$ (as vector space over field $k=\mathbb{Q}$). Thanks.

Find irreducible but not prime element in $\mathbb{Z}$

I have tried various numbers of the form $a+b\sqrt{5},\ a,b \in \mathbb{Z}$, but cannot find the one needed. I would appreciate any help. Update: I have found that $q=1+\sqrt{5}$ is irreducible. Now if I show that 2 is not divisible by $q$ in $\mathbb{Z}[\sqrt{5}]$ then $2\cdot2 = (\sqrt{5}-1)(\sqrt{5}+1)$ and I’m done. Can it be shown […]

Show that an algebraically closed field must be infinite.

Show that an algebraically closed field must be infinite. Answer If F is a finite field with elements $a_1, … , a_n$ the polynomial $f(X)=1 + \prod_{i=1}^n (X – a_i)$ has no root in F, so F cannot be algebraically closed. My Question Could we not use the same argument if F was countably infinite? […]

Show that $(\mathbb{Z}/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$

Show that $(\mathbb{Z}[x]/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$. Anyway how and what method is used to prove this. I still have no idea now. really thanks for your help

A counterexample to the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$

I am going over some counterexamples for the the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$. In particular I have been trying to understand what happens if you remove the various restrictions on the module conditions on $M$ necessary guarantee the bijectivity of the isomorphism. The strongest condition required that I know of $M$ be […]

Can the square of a proper ideal be equal to the ideal?

Let $R$ be a ring, commutative with $1$, let $\mathfrak{i}$ be an ideal, not the whole ring. In general $\mathfrak{i}^2\subseteq\mathfrak{i}$. Can this inclusion be an equality, or it is always a strict inclusion?