Galois Theory HT25, Misc
Flashcards
Rational root test
@State the rational root test.
Suppose:
- $a _ n x^n + a _ {n-1} x^{n-1} + \cdots + a _ 0 = 0$
- $a _ i \in \mathbb Z$, $a _ n \ne 0$
Then for any rational root $x = p/q$ fully reduced:
- $p$ divides $a _ 0$
- $q$ divides $a _ n$
@Justify that
\[2x^3 + x - 1\]
has no rational roots.
By the rational root test, any rational root $x =p/q$ fully reduced must have $p \mid 1$ and $q \mid 2$, so the only options are $x = \pm 1$ or $x = \pm \frac{1}{2}$.
@example~
@Justify that
\[x^3 + 5x + 1\]
has no rational roots.
By the rational root test, any rational root $x = p/q$ fully reduced must have $p \mid 1$ and $q \mid 1$, but as $x = 1$ is not a root, it has no rational root.
@example~
Useful irreducibility criterion
@State a useful criterion for when a polynomial $f \in F[t]$ is irreducible.
$f$ is irreducible iff $\text{Gal} _ F(f)$ acts transitively on the roots of $f$.
(E.g. useful if you know the Frobenius endomorphism is in the Galois group).
Examples of fields of rational functions
Let $L = \mathbb Q(x)$ be the field of fractions of $\mathbb Q[x]$. @Prove that any automorphism $\sigma : L \to L$ must satisfy
\[\sigma(x) = \frac{ax + b}{cx + d}\]
where $ad - bc \ne 0$.
Todo.