Undergrad Stewart's Galois Theory doesn't make sense

[swampwiz]

I am going through this book, and on page 38, there is

LEMMA 3.15
Let K be a subfield of C, f an irreducible polynomial over K, and g, h polynomials over K. If g divides gh, then either f divides h or f divides h.

OK, so I have proven that f must divide over g or h - i.e., if f doesn't divide g, it must divide h - but it seems that f could still divide both, which is not what the text says.

f = ( x - 1 )

g = ( x - 1 )² ( x - 2 )

h = ( x - 1 )³ ( x - 3 )

g h = ( x - 1 )⁵ ( x - 2 ) ( x - 3 )

Clearly, f divides ( g h ), g & h, so the LEMMA is wrong.

What am I missing here?

 

[WWGD]

I assume it is f divides gh. But the either is not necessarily exclusive. But, yes, it could be made more clear, I agree.

 

[fresh_42]

I am going through this book, and on page 38, there is

LEMMA 3.15
Let K be a subfield of C, f an irreducible polynomial over K, and g, h polynomials over K.
If g

f

divides gh, then either f divides h or f divides h.

g

This is simply the fact that irreducibility and primality are the same thing in $K[x]$. Lemma 3.15 if written correctly says, that any irreducible polynomial is prime.

 

[mathwonk]

in mathematics the phrase "either A or B" always means "either A or B or both".