I need to find the gcd of two polynomials: $f(x) = x^4+1$ and $g(x)=x^2-1$ using the Euclidean algorithm.
Wolfram shows that the gcd is equal to $1$, but for some reason I don't get the same answer.
What am I doing wrong?
If your polynomials are over $\mathbf{R}$, then a gcd of $1$ is equivalent to a gcd of $2$ (and any other nonzero real), because in general gcds are only well-defined up to associatedness, i.e. mutual divisibility.
The greatest common divisor of $f$ and $g$ is (in the case of polynomials) defined as a polynomial $d$ such that $d$ divides $f$ and $g$ and every divisor of $f$ and $g$ also divides $d$.
Thus, if we have another polynomial $e$ that is associated with $d$ (which means that $e|d$ and $d|e$), then we have $e|d|f$, $e|d|g$ and for every common divisor $z$ of $f$ and $g$ we have $z|d|e$, so $e$ is also a gcd of $f$ and $g$.
In your case you have two gcds of $1$ and $2$. Because $1\cdot 2 = 2$ and $2\cdot\frac12=1$ you have $1|2$ and $2|1$, so $1$ and $2$ are associated, thus if $1$ is a gcd, so is $2$ and vice versa.
Note that it is possible to normalize the gcd of polynomials by requiring the first nonzero coefficient to be $1$, which is what Wolfram|Alpha presumably does.
Note that gcds are preserved under scaling by units (invertibles) because this holds true for divisibility, i.e. if $\,u\,$ is a unit then $\,ua\mid b\iff a\mid b\iff a\mid ub.\,$ Thus scaling a gcd by a unit does not alter its multiples nor its divisors, so it remains a gcd (i.e. a common divisor that is greatest divisibility-wise, i.e. divisible by every common divisor; said equivalently $\, c\mid a,b \!\iff\! c\mid \gcd(a,b),\, $ the universal definition of a gcd).
Often it is convenient to scale gcds by a unit to a normal form, e.g. in $\,\Bbb Z\,$ we normalize gcds $\ge 0,\,$ by scaling by the unit $-1$ if need be, and in a polynomial ring $\,K[x]\,$ over a field, we normalize them to be monic (lead coeff $\,c_n = 1),\,$ by scaling the polynomial by $\,c_n^{-1}\,$ if need be (thus a constant gcd $\,c_0\neq 0$ normalizes to $1).\,$
However, a nice unit normalization algorithm need not exist in all domains, so generally gcds are only determined up to unit multiples, i.e. up to associate-ness, so e.g. $\,\gcd(a,b)\approx 1$ means that the gcd is associate to $1$, i.e. is a unit, i.e $\,c\mid a,b\iff c\mid 1.\,$ Beware that a common abuse of language is to write $\gcd(a,b) = c\,$ to denote $\,\gcd(a,b)\approx c,\,$ esp. in contexts when there is no natural choice for unit normalization.
See here for further discussion.
If you apply the Euclidean algorithm to $p(x)$ and $q(x)$ and the last non zero remainder is $44$, then there are polynomials $\alpha(x)$ and $\beta(x)$ such that $\alpha(x)p(x)+\beta(x)q(x)=44$. But then $\frac{\alpha(x)}{44}p(x)+\frac{\beta(x)}{44}q(x)=1$. So, $1$ is also a greatest common divisor of $p(x)$ and $q(x)$. Actually, if $r(x)$ is a greatest common divisor of $p(x)$ and $q(x)$, then so is $\frac{r(x)}k$, where $k$ is the coefficient of the monomial with highest exponent in $r(x)$. Then $\frac{r(x)}k$ is a monic polynomial, and it is usual to work with a monic polynomial when we are working with greatest common divisors of two polynomials.
The general principle behind this is the concept of units in a ring (with a $1$-element): https://en.wikipedia.org/wiki/Unit_(ring_theory)
A unit is an element of the ring that divides $1$ (the multiplicative neutral element). When you are dealing only with multiplication (like in this case), elements that differ only by a unit have exactly the same behaviour regarding divisibility (I assume a commutative ring here):
If $u$ is a unit and $x=yu$, then $x$ and $y$ divide exactly the same ring elements:
$$x|a \iff \exists f:fx=a \Rightarrow (fu)y=a \Rightarrow y|a$$
and with $u'={1\over u}$ we have $y=xu'$ and we get similiarly
$$y|a \iff \exists f:fy=a \Rightarrow (fu')x=a \Rightarrow x|a.$$
One can show very similiarly, that exactly the same ring elements divide $x$ and $y$.
So to state this again, regarding divisibilty, $x$ and $y$ have exactly the same properties.
That means if you find by some calulation that $\gcd(a,b)=x$, then you can also say $\gcd(a,b)=y$ (always assuming that $x=yu$ with $u$ being a unit). That means the $\gcd$ is only determined up to a unit!
When starting on these topics of divisibility and $\gcd$ etc. with $\mathbb Z$, this is usually ignored, as the only units of $\mathbb Z$ are $1$ and $-1$, and the general presentation on these topics is centered on positive integers, so the $-1$ gets ignored.
To come back to your problem, the ring in questions is ${\mathbb Q[x]}$. It's easy to check that in that ring the units are exactly the non-zero constant polynomials. That means that any $\gcd$ is only determined up to a unit (=constant polynomial). It is just as valid to claim the $\gcd$ equals the polynomial $p(x)=1$ or $q(x)=15$ or $r(x)=-2019.0319$, or whatever other constant comes to your mind.
As the other answers said, using the polynomial with the leading coefficient equal to $1$ is a general convention so results can be easily compared. It's the same (a convention) as writing one half usually as ${1 \over 2}$ and not the equally valid ${-17 \over -34}$.
