Minimal polynomial of the root of algebraic number I have obtained the minimal polynomial of $9-4\sqrt{2}$ over $\mathbb{Q}$ by algebraic operations:
$$ (x-9)^2-32 = x^2-18x+49.$$
I wonder how to calculate the minimal polynomial of $\sqrt{9-4\sqrt{2}}$ with the help of this sub-result? Or is there a smarter way to do this (not necessarily algorithmic)?

$\rm x^4-18\,x^2+49\:$ will be the minimal polynomial, unless $\rm\:\sqrt{9-4\sqrt{2}}\:$ denests to $\rm\:a + b\sqrt{2}.\:$
This can be tested by a radical denesting formula that I discovered as a teenager.

Simple Denesting Rule $\rm\ \ \ \color{blue}{subtract\ out}\ \sqrt{norm}\:,\ \ then\ \  \color{brown}{divide\ out}\ \sqrt{trace} $
Recall $\rm\: w = a + b\sqrt{n}\: $ has norm $\rm =\: w\:\cdot\: w' = (a + b\sqrt{n})\ \cdot\: (a - b\sqrt{n})\ =\: a^2 - n\: b^2 $  
and, furthermore,  $\rm\:w\:$ has trace $\rm\: =\: w+w' =  (a + b\sqrt{n}) + (a - b\sqrt{n})\: =\:  2\:a$

Here $\:9-4\sqrt{2}\:$ has norm $= 49.\:$ $\rm\ \: \color{blue}{subtracting\ out}\ \sqrt{norm}\ = 7\ $ yields $\  2-4\sqrt{2}\:$
and this has $\rm\ \sqrt{trace}\: =\: 2,\ \ so,\ \  \ \color{brown}{dividing\ it\ out}\ $ of this yields the sqrt: $\:1-2\sqrt{2}.$
Checking we have
$\ \smash[t]{\displaystyle \left(1-2\sqrt{2}\right)^2 =\ 1+8 -4\sqrt{-3}\ =\ 9-4 \sqrt{2}}.$
See this answer for general radical denesting algorithms.
