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Math Help - Field Extensions, Polynomial Rings and Eisenstein's Criterion

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    Super Member Bernhard's Avatar
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    Field Extensions, Polynomial Rings and Eisenstein's Criterion

    In Dummit and Foote Chapter 13: Field Theory, the authors give several examples of field extensions on page 515 - see attached.

    In example (3) we read (see attached)

    " (3) Take  F = \mathbb{Q} and  p(x) = x^2 - 2 , irreducible over  \mathbb{Q} by Eisenstein's Criterion, for example"

    Now Eisenstein's Criterion (see other attachment - Proposition 13 and Corollary14) require the polynomial to be in R[x] where R s an integral domain.

    In example (3) on page 515 of D&F we are dealing with a field, specifically  \mathbb{Q} .

    My problem is, then, how does Eisenstein's Criterion apply?

    Can anyone please clarify this situation for me?

    Peter
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    Re: Field Extensions, Polynomial Rings and Eisenstein's Criterion

    A field is an integral domain
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    Re: Field Extensions, Polynomial Rings and Eisenstein's Criterion

    Thanks Idea ... but see Deveno's post on Math Help Boards on this issue.

    Peter
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    Re: Field Extensions, Polynomial Rings and Eisenstein's Criterion

    I am not able to locate this

    'Deveno's post on Math Help Boards'

    Can you give me the link
    Thanks
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    Super Member Bernhard's Avatar
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    Re: Field Extensions, Polynomial Rings and Eisenstein's Criterion

    Hi Idea,

    Post by Deveno is post #8 at http:///linear-abstract-algebra-14/f...19-a-6555.html

    Part of Deveno's post reads as follows:

    1. When applying Eisenstein to rational polynomials, we typically turn them into integral polynomials by multiplying by the lcm of the denominators of the coefficients. This turns our polynomial into one over an integral domain, the integers. In the integers, we HAVE prime ideals, namely the ideals generated by a prime integer.

    Peter
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    Re: Field Extensions, Polynomial Rings and Eisenstein's Criterion

    If R is a ring, then $R[x]$ , the ring of polynomials in x with coefficients in R, consists of all formal sums $\displaystyle \sum_{i=0}^\infty a_i x^i$ , where $a_i = 0$ for all but finitely many values of i.

    If $\displaystyle \sum_{i=0}^\infty a_i x^i$ is a nonzero polynomial, the degree is the largest $n \ge 0$ such that $a_n \ne 0$ . The zero polynomial has degree $-\infty$ .

    $R[x]$ becomes a ring with the usual operations of polynomial addition and multiplication.

    If F is a field, the units in $F[x]$ are exactly the nonzero elements of F.

    If $F[x]$ is a field, $f(x), g(x) \in F[x]$ , and $g(x) \ne 0$ , there are unique polynomials $q(x), r(x) \in F[x]$ such that

    $$f(x) = q(x)\cdot g(x) + r(x), \quad\hbox{where}\quad \deg r < \deg g.$$ \item{$\bullet$} Let F be a field and let $f(x) \in F[x]$. c is a root of $f(x)$ in F if and only if $x - c \mid f(x)$.

    Let R be an integral domain. An element $x \in R$ is irreducible if $x \ne 0$ , x is not a unit, and if $x = yz$ implies either y is a unit or z is a unit.

    Let R be an integral domain. An element $x \in R$ is prime if $x \ne 0$ , x is not a unit, and $x \mid yz$ implies $x \mid y$ or $x \mid z$ .

    In an integral domain, primes are irreducible.

    Let F be a field. If $f(x), g(x) \in F[x]$ are not both zero, then $f(x)$ and $g(x)$ have a greatest common divisor which is unique up to multiplication by units (elements of F).

    Let F be a field, let $f(x), g(x) \in F[x]$ , and let $d(x)$ be a greatest common divisor of $f(x)$ and $g(x)$ . There are polynomials $u(x), v(x) \in F[x]$ such that

    $$d(x) = u(x)f(x) + v(x)g(x).$$


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    Re: Field Extensions, Polynomial Rings and Eisenstein's Criterion

    hello everyone
    welcome to this forum.Here u find all your questions and get your answers.
    right now i am unable to find your answer.If you want to see c programming you can go through this.
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    Re: Field Extensions, Polynomial Rings and Eisenstein's Criterion

    Hi James,

    Thanks for your post.

    I am finding it a little hard to read however. Are you able to correct your use of Latex. See the very helpful tutorials on the Latex Help forum

    Peter
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