# Thread:

1. ## splitting field L

B is a real number Sqrt(7 + 2*sqrt(7))

basically i have done the parts of finding the minimal polynomial mu of B over Q, and all the zeros of mu.

but i got stuck on the part where i need to find the degree [L :Q] and the basis for L over Q

Where L is the splitting field of mu over Q

can anyone help me find the degree and basis

thanks

2. Originally Posted by dopi
B is a real number Sqrt(7 + 2*sqrt(7))

basically i have done the parts of finding the minimal polynomial mu of B over Q, and all the zeros of mu.

but i got stuck on the part where i need to find the degree [L :Q] and the basis for L over Q

Where L is the splitting field of mu over Q

can anyone help me find the degree and basis

thanks
Let $x = \sqrt{7+2\sqrt{7}} \implies x^2 - 7 = 2\sqrt{7} \implies x^4 - 14x^2+21=0$.
Find the zeros of this polynomial and form the splitting field.

3. ## check solution

Originally Posted by ThePerfectHacker
Let $x = \sqrt{7+2\sqrt{7}} \implies x^2 - 7 = 2\sqrt{7} \implies x^4 - 14x^2+21=0$.
Find the zeros of this polynomial and form the splitting field.
i found the zeros as

$\beta = \pm \sqrt{7 \pm 2\sqrt{7}}$
$\alpha= \pm \sqrt{7 \mp 2\sqrt{7}}$

so i tried $\alpha\beta$, where i got $\sqrt{21}$, so is the splitting field L of mu over Q = $\sqrt{7}$.

is the degree[L,:Q] = 4x4 = 16
and the basis for L over Q { ${1, \sqrt{7},7, 7\sqrt{7}}$}
and the order of the galois group $\Gamma(L:Q) = 8$

4. Originally Posted by dopi
i found the zeros as

$\beta = \pm \sqrt{7 \pm 2\sqrt{7}}$
$\alpha= \pm \sqrt{7 \mp 2\sqrt{7}}$

so i tried $\alpha\beta$, where i got $\sqrt{21}$, so is the splitting field L of mu over Q = $\sqrt{7}$.

is the degree[L,:Q] = 4x4 = 16
and the basis for L over Q { ${1, \sqrt{7},7, 7\sqrt{7}}$}
and the order of the galois group $\Gamma(L:Q) = 8$
The splitting field is $\mathbb{Q}(\sqrt{7 + 2\sqrt{7}},\sqrt{7-2\sqrt{7}})$.

5. is the degree[L,:Q] = 4x4 = 16
and the basis for L over Q { ${1, \sqrt{7},7, 7\sqrt{7}}$}
and the order of the galois group $\Gamma(L:Q) = 8$[/QUOTE]

i was wondering are these correct?
{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "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": 0.8858636617660522, "perplexity_flag": "middle"}