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Math Help - UFDs, primes and irreducibles - Dummit and Foote - Chapter 8

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    Super Member Bernhard's Avatar
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    UFDs, primes and irreducibles - Dummit and Foote - Chapter 8

    I am reading Dummit and Foote Chapter 8, Section 8.3 UFDs.

    On page 284 (see attached) Dummit and Foote prove Proposition 10 which shows the following:

    "In an integral domain a prime element is always irreducible."

    D&F then state:

    "It is not true in general that an irreducible element is necessarily prime."

    They give as an example the element 3 in the quadratic integer ring  R =\mathbb{Z} [\surd -5 ]

    They assert that the element 3 is irreducible but not prime.

    I am struggling to show rigorously that 3 is irreducible in R, despite D&F's reference to the calculations on page 273 (see attached)

    Can someone please help me produce a formal and rigorous demonstration that 3 is irreducible.

    That is to show that whenever  3 = ( a_1 + b_1 \surd -5 ) (a_2 + b_2 \surd -5 )

    then one of  ( a_1 + b_1 \surd -5 ) , (a_2 + b_2 \surd -5 ) must be a unit in  \mathbb{Z} [\surd -5 ]


    Peter
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    Last edited by Bernhard; October 25th 2012 at 11:53 PM.
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    Re: UFDs, primes and irreducibles - Dummit and Foote - Chapter 8

    Quote Originally Posted by Bernhard View Post
    "It is not true in general that an irreducible element is necessarily prime."

    They give as an example the element 3 in the quadratic integer ring  R =\mathbb{Z} [\surd -5 ]

    They assert that the element 3 is irreducible but not prime.

    I am struggling to show rigorously that 3 is irreducible in R, despite D&F's reference to the calculations on page 273 (see attached)

    Can someone please help me produce a formal and rigorous demonstration that 3 is irreducible.
    You've had this shown to you, that 3 is irreducible in this ring, at least twice in the last month. Look them over again:

    Existence of Greatest Common Divisors - Dummit and Foote Ch8

    Euclidean Domains - Dummit and Foote - Chapter 8 - Section 8.1 - Example on Quadratic
    Last edited by johnsomeone; October 26th 2012 at 03:00 PM.
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    Super Member Bernhard's Avatar
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    Re: UFDs, primes and irreducibles - Dummit and Foote - Chapter 8

    Thanks for pointing that out ... And apologies for forgetting this ... My day job intrudes on my Maths to the point I sometimes have to revisit problems.

    Will now go to the posts you mention

    Thanks again,

    Peter
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    Re: UFDs, primes and irreducibles - Dummit and Foote - Chapter 8

    if 3 were reducible in Z[√(-5)], say 3 = ab where a,b are not units, we would have N(3) = 9 = N(ab) = N(a)N(b) (here N is the field norm, or the square of the complex modulus. this is always a positive integer).

    if N(a) OR N(b) = 1, then it must be 1, and both of these are units.

    hence N(a) = N(b) = 3.

    suppose a = c + d√(-5) where c,d are integers.

    N(a) = 3 implies c2+5d2 = 3.

    thus d = 0, (or else N(a) > 3) and c2 = 3 has no integer solution.
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    Super Member Bernhard's Avatar
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    Re: UFDs, primes and irreducibles - Dummit and Foote - Chapter 8

    Thanks Deveno ... Most helpful

    Peter
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