0. Facts true for all fields

If $F$ is any field, then $F[x]$ has essentially the same algebraic properties as $\mathbb{Z}$. In particular,

• $F[x]$ is a ring. (Recall that $F[x]$ with the square brackets is the set of all polynomials with coefficients in $F$. Be sure not to confuse $F[x]$ and $f(x)$.)

• The units of $F[x]$ are the nonzero constant polynomials.

• The degree of a polynomial is used as a kind of measure of its size, unlike the integers where the absolute value is a measure of size.

• If $a(x)$ and $f(x)$ are in $F[x]$ and $a(x) \neq 0$ then we can divide to get $f(x) = q(x) a(x) + r(x)$ for some quotient $q(x)$ and remainder $r(x)$ with deg$\; r(x) <$   deg$\; a(x)$.

• The Euclidean algorithm works to find the greatest common divisor of two polynomials.

  (The gcd is unique up to multiplying by a unit.)

• Bezout works: If $d(x) =$   gcd$(f(x), g(x))$ then there exist $u(x)$ and $w(x)$ with $d(x) = u(x) f(x) + w(x) g(x)$.

• If $f(x)$ is an irreducible polynomial and $f(x) \vert a(x) b(x)$ then $f(x) \vert a(x)$ or $f(x) \vert b(x)$.

• Factorization:
  • For polynomials that can't be factored into smaller ones, meaning into polynomials of lower degree, we say ``irreducible'' instead of ``prime'', but it's the same idea.

  • Every polynomial can be factored into irreducibles. This factorization is unique up to order and up to unit factors.

  • Usually we write a factorization as (say) $f(x) = c f _ 1(x)^{e _ 1} \dots f _ k(x)^{e _ k}$, where $c$ is a nonzero constant (i.e., a unit), and each of the $f _ i(x)$ is monic (has leading coefficient 1).

    Example: In $\mathbb{R}[x]$, $4x^5 + 8x^4 + 4x^3 = 4 x^3 (x+1)^2$.

• Roots:
  • To say that $r$ in $F$ is a root of $f(x)$ means that $f(r) = 0$. Notice that when you say ``root'' you don't need to write `` $= 0$''; we can just say ``$r$ is a root of $f(x)$'' instead of ``$r$ is a root of $f(x) = 0$'', although that's OK.

  • $r$ in $F$ is a root of $f(x)$ $\Leftrightarrow$ $(x-r) \vert f(x)$.

  • A polynomial $f(x)$ of degree $n$ can have at most $n$ roots.

    Reason: For each distinct root $r _ 1, r _ 2,\dots$, divide $f(x)$ by $(x - r _ 1)$, then by $(x - r _ 2)$, and so on; the degree of what is left keeps going down so no more than $n$ linear factors are possible.

  • It is possible for $f(x)$ of degree $n$ to have fewer than $n$ roots, for two reasons:

    (i) There might be a repeated linear factor such as $(x-1)^3$. Then the root 1 is called a ``multiple root'', or more specifically , a ``root of multiplicity 3''. We think of it as having the root 1 three times.

    (ii) There might be a nonlinear irreducible factor, such as $(x^2 + 1)$ in $\mathbb{R}[x]$.

  • An irreducible polynomial $f(x)$ in $F[x]$ of degree $> 1$ does not have any roots in $F$. (However, $f(x)$ could have a root in a larger field containing $F$. In fact, there is such a field.)

• Congruences:
  • Congruences in $F[x]$ modulo $f(x)$ work the way we are used to for integers.

  • Congruence classes (our ``boxes'') work and make a ring,
    $F[x]/f(x)F[x]$.

  • If $f(x)$ is irreducible, then $F[x]/f(x)F[x]$ is a field, just as $\mathbb{Z}/p\mathbb{Z}$ is a field if $p$ is prime.

    Example: $\mathbb{R}[x]/(x^2 + 1)\mathbb{R}[x]$ is a field.

  • In fact, if $f(x)$ is irreducible, then $F[x]/f(x)F[x]$ is a field containing $F$, in which $f(x)$ now has a root, namely the class (``box'') of $x$ modulo $f(x)$.

    Example: $\mathbb{R}[x]/(x^2 + 1)\mathbb{R}[x]$ is a field in which $x^2 + 1$ has a root, which we can call $i$; this field is isomorphic to $\mathbb{C}$. In fact, this is one way to construct $\mathbb{C}$.

    Note: There isn't time left in the course to do much with constructing fields from polynomial congruences, so questions on the final will be restricted to knowing examples from lecture and homework.

    Even so, do be aware of these larger ideas:

    • If $f(x)$ in $F[x]$ is irreducible and doesn't have a root in $F$, it is always possible to construct a larger field in which $f(x)$ does have a root. An example is that $x^2 + 1$ has no root in $\mathbb{R}$ but does in $\mathbb{C}$.

    • The idea of constructing new mathematical objects by using ``boxes'' (equivalence classes) occurs frequently in advanced mathematics courses.

    • The idea of regarding two objects as the same if they are isomorphic is also very common in advanced mathematics courses. So it doesn't matter whether we think of $\mathbb{C}$ as being made from $2 \times 2$ real matrices, or as being made from congruence classes, or as consisting of expressions $a + bi$ that are added and multiplied in a certain way. The first two ways assure us that $\mathbb{C}$ is possible and is a field; the last way is for everyday use.