What is the galois group of x+3 or (x+1)(x+2) ? How about A(x)B(x)?

jhenezhubby01ff 2022-10-01 Answered
What is the galois group of x + 3 or ( x + 1 ) ( x + 2 ) ? How about A ( x ) B ( x ) ?
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Answers (1)

Cameron Wallace
Answered 2022-10-02 Author has 8 answers
The idea that the Galois group of a product of polynomials is the direct product of the groups is false, however (assuming that is what you mean). Consider A ( x ) = B ( x ) = x 2 + 3. Then A ( x ) B ( x ) = ( x 2 + 3 ) 2 and this polynomial also splits completely over Q ( 3 ). So the Galois group of A ( x ) B ( x ) is still C 2 C 2 × C 2
For the composition consider A ( x ) = x 3. Then A ( B ( x ) ) = x 2 has trivial Galois group, but B has Galois group C 2
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