# Does this map extend to a unique automorphism of the polynomial ring of an integral domain?

Suppose that $A$ is an integral domain. If $a,b$ are element of $A$ and $a$ is a unit, then how come the change of variable $X\mapsto aX+b$ extends to a unique automorphism of the polynomial ring $A[X]$, which equals the identity map when restricted to $A$?

Thanks for any ideas or ways to get started.

-

## 1 Answer

Well, to start with the polynomial Ring $A[X]$ has the universal property, that for every morphism $f : A \rightarrow B$ of Rings and every $c \in B$ there is a unique homomorphism $\overline{f}: A[X] \rightarrow B$ which extends $f$ and maps $X$ to $c$. That is, if $p = \sum_{i = 1}^n a_i X^i$ is given, we have $$\overline{f}(p) = \sum_{i = 1}^n f(a_i) c^i.$$ In your case, you can take $B = A[X]$, $f: A \rightarrow A[X]$ the canonical inclusion and $c = a X + b$. Now the only thing left to be proved is that this is indeed a bijection. This can be done by explicitly stating the inverse and here you will need that $a$ is a unit.

-

Thank you. ${}{}$ – Clara Nov 22 '11 at 20:24
{"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": 20, "mathjax_display_tex": 1, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9130785465240479, "perplexity_flag": "head"}