What is the free algebra $A=k\langle X_1,...,X_n \rangle$? And why is it an algebra? I'm reading a book, where they claim that the free algebra $A=k\langle X_1,...,X_n\rangle$ is an algebra. I've never seen this notation and I've never heard of the free algebra, so I wonder how this is defined and why it is an algebra.
 A: The free algebra $k\langle X_1, \ldots, X_n\rangle$ is, according to wikipedia, the non-commutative analogue of the polynomial ring in $n$ variables. That means that the basis elements in the free algebra are words
$$
X_{i_1}X_{i_2}\cdots X_{i_m}
$$
and a general element in the algebra is a sum of finitely many such words, each with a coefficient from $k$. So, for instance, one element of $\Bbb R\langle x, y, z\rangle$ looks like this
$$
3yzy - 2xzyx + 5
$$
Multiplication is carried out like you usually would multiply two polynomials, expanding parentheses ("distributing") and so on, just with the added detail that the variables don't commute, so you cannot rearrange $xyx$ into $x^2y$, and you'll have to keep track of which element is the left side and which is the right side of a multiplication. You can still rearrange terms, though, so $x + y$ is the same as $y + x$.
Example multiplication:
Take the two elements $x -y$ and $x+y$ in $\Bbb R\langle x, y\rangle$. Their product is
$$
(x-y)(x+y)=x(x+y) - y(x + y)\\
= xx + xy - yx - yy\\
= x^2 + xy - yx - y^2
$$
and since it's non-commutative, $xy$ and $yx$ are completely different terms, and thus this cannot be simplified any further.
A: It's by definition an algebra, as you define it to be the algebra generated by $k$ and $n$ algebraically-independent transcendental variables $X_1,\ldots,X_n$.
