- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

I am reading Paul E Bland's book: The Basics of Abstract Algebra and I am trying to understand his definition of "separable polynomial" and his second example ...

Bland defines a separable polynomial as follows:

... and Bland's second example is as follows:

I am uncomfortable with, and do not fully understand the above definition and am uncomfortable with the example as well ... I hope someone can clarify my difficulties and problems ...

The first and second sentences of the definition above seem to lead to different notions of separability to me ...

The first sentence of the definition:

" ... ... For a field ##F##, an irreducible polynomial ##f(x) \in F[x]## of degree ##n## is said to be separable if ##f(x)## has ##n## distinct roots in its splitting field ... ... "

Under this definition of a separable polynomial, the polynomial in Bland's example:

##f(x) = (x^2 + 2)^2 (x^2 - 3)##

is of degree ##n = 6## and splits in (among other fields) in ##\mathbb{C}## and does NOT have ##6## distinct roots (as the roots ##\pm \sqrt{2} i## are repeated) ... ...(EDIT ... hmm ... last minute thought! ... but i guess you could argue that ##f(x)## is not irreducible ... is that the key to my confusion ...?)... BUT ...The second sentence of Bland's definition reads:

" ... ... A polynomial in ##F[x]## is said to be separable if each of its irreducible factors is separable if each of its irreducible factors is separable ... "

Well ... under this definition I (uncomfortably) go along with Bland's analysis of

##f(x) = (x^2 + 2) (x^2 + 2) (x^2 - 3)##

in his example ...BUT i remain uncomfortable with this ... I cannot think of an example of a case where this definition gives rise to a repeated root ...can someone please give a simple example of a polynomial with a repeated root under Bland's definition ...

Help will be much appreciated ...

Peter

Bland defines a separable polynomial as follows:

... and Bland's second example is as follows:

I am uncomfortable with, and do not fully understand the above definition and am uncomfortable with the example as well ... I hope someone can clarify my difficulties and problems ...

The first and second sentences of the definition above seem to lead to different notions of separability to me ...

The first sentence of the definition:

" ... ... For a field ##F##, an irreducible polynomial ##f(x) \in F[x]## of degree ##n## is said to be separable if ##f(x)## has ##n## distinct roots in its splitting field ... ... "

Under this definition of a separable polynomial, the polynomial in Bland's example:

##f(x) = (x^2 + 2)^2 (x^2 - 3)##

is of degree ##n = 6## and splits in (among other fields) in ##\mathbb{C}## and does NOT have ##6## distinct roots (as the roots ##\pm \sqrt{2} i## are repeated) ... ...(EDIT ... hmm ... last minute thought! ... but i guess you could argue that ##f(x)## is not irreducible ... is that the key to my confusion ...?)... BUT ...The second sentence of Bland's definition reads:

" ... ... A polynomial in ##F[x]## is said to be separable if each of its irreducible factors is separable if each of its irreducible factors is separable ... "

Well ... under this definition I (uncomfortably) go along with Bland's analysis of

##f(x) = (x^2 + 2) (x^2 + 2) (x^2 - 3)##

in his example ...BUT i remain uncomfortable with this ... I cannot think of an example of a case where this definition gives rise to a repeated root ...can someone please give a simple example of a polynomial with a repeated root under Bland's definition ...

Help will be much appreciated ...

Peter