# Is $t^{2}$ a prime element of $\mathbb{F}_{2}(t^{2},s^{2})$?

I wish to find out if $t^{2}$ is a prime element of $\mathbb{F}_{2}(t^{2},s^{2})$ so I can justify the use of Eisenstein on the polynomial $x^{2}-t^{2}\in\mathbb{F}_{2}(t^{2},s^{2})[x]$

I believe that it does since $\mathbb{F}_{2}(t^{2},s^{2})/\langle t^{2}\rangle\cong\mathbb{F}_{2}(s^{2})$ is an integral domain (since it is a field)

Is my argument correct and I may use Eisenstein ? (I already noted that $t^{4}$ does not divide $t^{2}$)

-

1
$\mathbb{F}_2(t^2, s^2)$ is a field. It has no prime elements, and the ideal generated by $t^2$ is the whole thing, so the quotient is zero. – Qiaochu Yuan Oct 11 '12 at 5:24
@QiaochuYuan - I see, thank you. It seems that if I replace the brackets with $[ ]$ then it is a prime elemnt. Can I justify using Eisenstein now ? – Belgi Oct 11 '12 at 5:26
Yes, but you don't need to. You can give a direct proof. – Qiaochu Yuan Oct 11 '12 at 5:38
@QiaochuYuan If you can think of such an argument I would love to hear it, please take the time to write it. Its about the principle (I 'feel' Eisenstein and I can't argue the conditions). The question itself is not as important. if I can't seem to think of how to justify the use of Eisenstein (and you say it can be justified) then I either missing something important or not understanding some concept and this troubles me – Belgi Oct 11 '12 at 5:43

## 2 Answers

Two comments.

1. A general form of Eisenstein's criterion shows that $x^2 - t^2$ is irreducible in $\mathbb{F}_2[t^2, s^2][x]$. Since $\mathbb{F}_2[t^2, s^2]$ is a UFD, a general form of Gauss's lemma shows that $x^2 - t^2$ is irreducible in $\mathbb{F}_2(t^2, s^2)[x]$, and primeness is equivalent to irreducibility in a UFD.

2. Neither of the above general facts are necessary in this case. If you only want to show primeness in $\mathbb{F}_2[t^2, s^2][x]$, it suffices to observe that the quotient by $x^2 - t^2$ is $\mathbb{F}_2[s^2, x]$, which is an integral domain. If you want to show primeness in $\mathbb{F}_2(t^2, s^2)[x]$, you can observe directly that the only form a nontrivial factorization can take is $(x - a)(x + a)$ for some $a \in \mathbb{F}_2(t^2, s^2)$, and by degree considerations no such $a$ satisfies $a^2 = t^2$. This shows irreducibility and, again, irreducible is equivalent to prime in a UFD.

-

Can you please add explanation about the use of Eisenstein ? Where exactly do I need $t^2$ to be a prime element, if I say that $\mathbb{F}_{2}[t^{2},s^{2}][x]/\langle t^{2}\rangle\cong\mathbb{F}_{2}[s^{2}]$ then I have what I need ? I really appriciate your help! – Belgi Oct 11 '12 at 5:58
@Belgi: I don't really understand the question. – Qiaochu Yuan Oct 11 '12 at 6:00
I will try to be more clear: I wish to use Eisenstein in its general form to show that $x^{2}-t^{2}\in\mathbb{F}_{2}(t^{2},s^{2})[x]$ is irreducible . I believe we have shown $x^{2}-t^{2}\in\mathbb{F}_{2}[t^{2},s^{2}][x]$ using Eisenstein with the integral domain $R=\mathbb{F}_{2}[t^{2},s^{2}]$ and the prime ideal $\langle t\rangle$ of $R$. But I want $x^{2}-t^{2}$ to be irreducible in $\mathbb{F}_{2}(t^{2},s^{2})[x]$ not just in $\mathbb{F}_{2}[t^{2},s^{2}]$. – Belgi Oct 11 '12 at 6:08
– Qiaochu Yuan Oct 11 '12 at 6:13
Oh, So I have to use both Eisnstien and Gauss's lemma and not only Eisnstien , right ? thank you for the help (+1) – Belgi Oct 11 '12 at 6:15
show 1 more comment

Hint $\$ prime $\rm t\in R[t]\iff$ domain $\rm\:R[t]/(t)\cong R$

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 37, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9331730008125305, "perplexity_flag": "head"}