The definition of 'reducible' in Hungerford's Algebra text

In summary, the term 'reducible' is used in a textbook to refer to a polynomial that can be factored into two elements that are not units. It is also used in the context of a proof and in a proposition to mean 'not irreducible'. The author questions whether this interpretation is safe in the text.
  • #1
julypraise
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He starts using the term 'reducible', as it came out of nowhere, from the page 162 of the text.

I know, roughly, what kind of thing he mean by this 'reducible' obejct. (That is that an element is factored into two elements that are not units.) And this should not be a problem if this term is used in only a informal essay type discussion level. But then, on the page 164, he uses this term in the proof (thm 6.13). And also, the page 273, the proof of the prop 4.11: 'f is either irreducible or reducible', which kinda suggests that 'reducible' means 'not irreducible'. Also page 274, on the top and the bottom both.

This is a problem for me.

If I take 'reducible' as 'not reducible' then 0 and units are reducible too, which may be a problem. But it maybe not at least in this text.

So is this interpretation safe in this text?



Please help me, someone who know this textbook quite well. Thanks.
 
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  • #2
In all those instances he's referring to a polynomial, where the adjective "reducible" has its usual meaning.

I guess generally one would call a nonzero nonunit reducible if it isn't irreducible - equivalently, if you could write it as a product of two nonzero nonunits.
 
  • #3
Okay, I will take note of that.
 

1. What is the definition of 'reducible' in Hungerford's Algebra text?

In Hungerford's Algebra text, reducible refers to a polynomial that can be factored into two or more non-constant polynomials with coefficients in the same field.

2. How is 'reducible' different from 'irreducible' in Hungerford's Algebra text?

The term 'irreducible' in Hungerford's Algebra text refers to a polynomial that cannot be factored into two or more non-constant polynomials with coefficients in the same field. In other words, an irreducible polynomial is a prime polynomial that cannot be broken down further.

3. What does it mean for a polynomial to be 'reducible by a linear factor'?

A polynomial is reducible by a linear factor if it can be factored into the product of a linear polynomial and another polynomial with lower degree.

4. Can a polynomial be both 'reducible' and 'irreducible'?

No, a polynomial cannot be both reducible and irreducible. A polynomial is either one or the other, depending on whether or not it can be factored into non-constant polynomials with coefficients in the same field.

5. How do you determine if a polynomial is 'reducible' or 'irreducible' in Hungerford's Algebra text?

To determine if a polynomial is reducible or irreducible in Hungerford's Algebra text, you must factor the polynomial and check if it can be broken down further. If it can be factored into non-constant polynomials, then it is reducible. If it cannot be factored any further, it is irreducible.

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