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§ 8.2 Multiplicity of Roots

 
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 - r₁)^m₁ (x - r₂)^m₂ · · · (x - r_t)^m_t

 
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,

 
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) = a₀ + a₁ x^p + a₂ x^2p + · · · + a_n x^np.

 
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.

 
Theorem 8.2.8. Let F be a finite extension of the field K. If F is separable over K, then it is a simple extension of K.

§ 8.2 Solved problems

 
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 x⁶-1 over Z₇. Solution

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