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Definition 8.2.1.
Let f(x) be a polynomial in K[x],
and let F be a splitting field for f(x) over K.
If f(x) has the factorization
f(x) = (x - r_{1})^{m1} (x - r_{2})^{m2} · · · (x - r_{t})^{mt}
over F, then we say that the root r_{i} has multiplicity m_{i}.
Definition 8.2.2.
Let f(x) be a polynomial in K[x], with f(x) =
a_{k} x^{k}. The
formal derivative
f'(x) of f(x) is defined by the formula
f'(x) = k a_{k} x^{k}^{-1},
where k a_{k} denotes the sum of a_{k} added to itself k times.
Proposition 8.2.3.
The polynomial f(x) in K[x]
has no multiple roots if and only if
gcd(f(x),f'(x)) = 1.
Proposition 8.2.4.
Let f(x) be an irreducible polynomial over the field K.
Then f(x) has no multiple roots unless
chr(K) = p 0
and f(x) has the form
f(x) = a_{0} + a_{1} x^{p} + a_{2} x^{2}^{p} + · · · + a_{n} x^{n}^{p}.
Definition 8.2.5.
A polynomial f(x) over the field K is called
separable
if its irreducible factors have only simple roots.
An algebraic extension field F of K is called
separable over K
if the minimal polynomial of each element of F is separable.
The field F is called
perfect
if every polynomial over F is separable.
Theorem 8.2.6.
Any field of characteristic zero is perfect.
A field of characteristic p>0 is perfect
if and only if each of its elements has a pth root.
Corollary 8.2.7.
Any finite field is perfect.
8.
Let f(x) be a polynomial in Q[x]
be irreducible over Q,
and let F be the splitting field for f(x) over Q.
If [F:Q] is odd, prove that all of the roots of f(x) are real.
Solution
9. Find an element a with Q( , i) = Q(a). Solution
10. Find the Galois group of x^{6}-1 over Z_{7}. Solution
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