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I'm preparing assignment questions for a course in ring/field theory. We'll shortly be looking at extension fields, and the students are meant to understand what notation such as $\mathbb{R}(i)$ means.

Two examples of questions that might arise:

  • If the question asked to describe the elements in $\mathbb{Z}_5(\sqrt{3})$, the student would be expected to recognise that the polynomial $x^2-3$ has no root in $\mathbb{Z}_5$, and thus is irreducible in $\mathbb{Z}_5[x]$, and thus define some element $\omega$ as a root of $x^2-3$, and set $\mathbb{Z}_5(\sqrt{3})=\mathbb{Z}_5(\omega)$, from which they can identify the elements.

  • If the question asked to describe the elements in $\mathbb{Z}_7(\sqrt{2})$, the student would be expected to recognise that the polynomial $x^2-2=(x-3)(x-4)$, and thus is reducible in $\mathbb{Z}_7[x]$, and thus recognise that $\mathbb{Z}_7(\sqrt{2})=\mathbb{Z}_7$ (via: if we define $\omega$ as a root of $x^2-2$ then $\omega \in \{3,4\}$, and so $\omega \in \mathbb{Z}_7$).

However, it is possible to have a "partially reducible" polynomial (i.e., one that does not factor entirely into linear factors in $F[x]$). Which leads to the following question:

Does $\mathbb{Z}_5(\sqrt[3]{3})$ make sense? (I.e. The extension field containing $\mathbb{Z}_5$ and $\sqrt[3]{3}$.)

If we use the above approach, we identify that the polynomial $x^3-3=(x-2)(x^2+2x+4)$, and thus is reducible in $\mathbb{Z}_5[x]$. If we define $\omega$ as a root of the polynomial $x^3-3$, then:

  • $\omega$ might be $2$, in which case $\omega \in \mathbb{Z}_5$ and $\mathbb{Z}_5(\omega)=\mathbb{Z}_5$.
  • $\omega$ might instead be a root of $x^2+2x+4$, in which case $\mathbb{Z}_5(\omega) \neq \mathbb{Z}_5$.

In order to reconcile this problem, it looks like there should be some restriction placed on the notation $F(\omega)$, such as "this is semantically incorrect, unless $\omega$ is derived from an irreducible polynomial in $F[x]$", but this would ruin the second question above.

  • 2
    Most of what you write is irrelevant to your question, no? You are simply asking if the notation $\sqrt[3]{3}$ is ambiguous. And yes, it is :-) Over the real field there are certain standard conventions used, but they do not apply muchly to other fields; and even then, $\sqrt[60]{1}$ is probably something one should not write even when talking about $\mathbb C$ without making a specific local convention.2012-09-10
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    You can talk about the ring $\mathbb{F}_5 [x]/(x^3 - 3)$, but it is not a field.2012-09-10

2 Answers 2