Why is $L=\mathbb{Q}(\sqrt[1]{2})\cup\mathbb{Q}(\sqrt[2]{2})\cup\mathbb{Q}(\sqrt[3]{2})\cup\cdots$ a field?

The title sais it already:

Why is $L=\mathbb{Q}(\sqrt[1]{2})\cup\mathbb{Q}(\sqrt[2]{2})\cup\mathbb{Q}(\sqrt[3]{2})\cup\cdots$ a field?

The hint provided in my textbook is: $\mathbb{Q}(\sqrt[n]{2})\cup\mathbb{Q}(\sqrt[m]{2})\subset\mathbb{Q}(\sqrt[mn]{2})$, but this doesn't really get me anywhere. Actually, I have no idea what to do whatsoever. Could anyone clarify or give some hint please?

• Hint: Suppose $\alpha^2=2, \beta^3 = 2$. What is $(\alpha \beta)^6$? Can you use that to find $\gamma$ in your union such that $\gamma^6=2$? – lulu Mar 17 '17 at 0:54
• Have you tried verifying each of the axioms of a field one by one? Which ones are giving you trouble? – Steven Stadnicki Mar 17 '17 at 0:56
• That's because it's a direct limit of fields – Bernard Mar 17 '17 at 0:57
• @Tyron don't try to avoid what seems like the more tedious method. If you can't explain why that union is closed under addition, for example, then you really owe it to yourself to figure that out. Later you may understand more slick solutions, but it's better to have a solution than to have none at all. – KCd Mar 17 '17 at 2:00
• More generally, if $\{L_i:i\in I\}$ is any collection of fields indexed by $I$ with the property that for all $i,j\in I$ there exists a $k\in I$ such that $L_i,L_j\subset L_k$, then $L=\bigcup_{i\in I}L_i$ is a field. It is actually easy to verify it is closed under addition, multiplication, inverses etc. Suppose $a,b\in L$... (go from there) – arctic tern Mar 17 '17 at 3:01

2 Answers

Suppose you take any two elements $x, y\in L.$ This means that there are some positive integers $m$ and $n$ such that $x\in \mathbb{Q}(\sqrt[m]{2})$ and $y\in \mathbb{Q}(\sqrt[n]{2})$.

From the hint in your textbook (that you wrote in your question), this implies that $x,y\in \mathbb{Q}(\sqrt[mn]{2}).$

Now, your end goal is to show that $x+y$ and $xy$ are both elements of $L$ as well (and that $L$ is closed under inverses for nonzero elements).

Is there any information about $\mathbb{Q}(\sqrt[mn]{2})$ that you can use which might help here (since $x$ and $y$ are elements of this set)?

• Very economical. +1. – Lubin Mar 17 '17 at 16:15

The smallest field containing $K(a^{1/n})$ and $K(a^{1/m})$ is $K(a^{1/n},a^{1/m}) = K(a^{1/lcm(n,m)})$

• Hello, thank you for your reply. Unfortunately, I do not fully understand what you are getting at. It seems to me that you are proofing the hint my textbook provided, but my guess would be that your statement contains more information than just that. Could you please elaborate? – Tyron Mar 17 '17 at 0:55
• $K(a^{1/n},a^{1/m}) \subset K(a^{1/lcm(n,m)})$ is obvious. What you need is proving the other direction @Tyron – reuns Mar 17 '17 at 0:58