## Return to Answer

Post Undeleted by D. Savitt
2 deleted 255 characters in body; added 8 characters in body

Here's a strategy that I've been toying with. It seems unlikely to me that it works, but perhaps it'll inspire someone to have a better idea. Fix an embedding of $\overline{\mathbb{Q}}$ into $\mathbb{C}$, thereby fixing Try to arrange a complex conjugation $c$. Let $L$ be the fixed field of $\overline{\mathbb{Q}}$ under a Sylow $3$-subgroup of the absolute Galois group of $\mathbb{Q}$, and let $K$ be the fixed field of $L$ under $c$. Then the whose Galois group of $K$ is an extension of has trivial pro-$5$ Sylow subgroup but nontrivial pro-$p$ Sylow subgroup for $\mathbb{Z}/2\mathbb{Z}$ by a big pro-$3$ group. p=2,3$. Let$f$be the product of an irreducible quadratic and an irreducible cubic over$K$. Since there are no irreducible quintics over$K$, for each$a \in K$, the polynomial$f-a\$ either has a root, or factors as an irreducible quadratic times an irreducible cubic. We'd be done if we can arrange for the latter to occur only finitely many times. But this seems like a stretch....

Post Deleted by D. Savitt
Post Undeleted by D. Savitt
Post Deleted by D. Savitt
1

Here's a strategy that I've been toying with. It seems unlikely to me that it works, but perhaps it'll inspire someone to have a better idea. Fix an embedding of $\overline{\mathbb{Q}}$ into $\mathbb{C}$, thereby fixing a complex conjugation $c$. Let $L$ be the fixed field of $\overline{\mathbb{Q}}$ under a Sylow $3$-subgroup of the absolute Galois group of $\mathbb{Q}$, and let $K$ be the fixed field of $L$ under $c$. Then the Galois group of $K$ is an extension of $\mathbb{Z}/2\mathbb{Z}$ by a big pro-$3$ group. Let $f$ be the product of an irreducible quadratic and an irreducible cubic over $K$. Since there are no irreducible quintics over $K$, for each $a \in K$, the polynomial $f-a$ either has a root, or factors as an irreducible quadratic times an irreducible cubic. We'd be done if we can arrange for the latter to occur only finitely many times. But this seems like a stretch....
{"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": 38, "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": 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": 0.960721492767334, "perplexity_flag": "head"}