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Hi, I'm looking at proposition 1.14(c) of Artin's Algebra.

It says

if we have K a splitting field for polynomial f from F[x], with roots a_1,...,a_n,

then the Galois group G(K/F) acts faithfully on the set of roots.

I look at faithful as the symmetries in the roots completely represent the group.

That is, no root is fixed by any group element (besides the identity (edited)).

When should I worry about this, are there any ways to construct a relevant counterexample if we drop a condition?

(So for actions, transitive and faithful is like surjectivity and injectivity respectively?)

It says

if we have K a splitting field for polynomial f from F[x], with roots a_1,...,a_n,

then the Galois group G(K/F) acts faithfully on the set of roots.

I look at faithful as the symmetries in the roots completely represent the group.

That is, no root is fixed by any group element (besides the identity (edited)).

When should I worry about this, are there any ways to construct a relevant counterexample if we drop a condition?

(So for actions, transitive and faithful is like surjectivity and injectivity respectively?)

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