Beginning questions in number theory

I have just a couple of quick questions about the lecture yesterday. In the notes you have lectures 1 to 3, and are these the notes for the 3 hour lecture yesterday?
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The material is more or less the same, but I am presenting the material in different order. Eventually, I will put up a new set where the order is more consistent with the lectures.

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Also, I am a bit unclear about the link between roots, factors and reducibility of polynomials in Q[x]. Does reducibility basically mean existence of roots and vice versa?

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A non-constant polynomial f \in Q[x] is reducible if it can be written as f=gh where both g and h have degree strictly less than the degree of f. This will be true if f has a root a\in Q, since then f(x)=(x-a)h(x) for some h. However, this isn’t the only way to be reducible. For example,

(x^2+1)(x^2-2)

is clearly reducible in Q[x], but it has no root.

That is, if f is irreduble, then it has no root. But the converse is false in general.

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For degree 2 or 3 polynomials p(x) in Q[x] you said that p(x) is irreducible if it has no roots in Q. Is this also true for higher degree? We know by Fundamental Theorem of Algebra that any nonconstant polynomial in C[x] has roots in C, and therefore the only irreducible polynomials in C[x] are those which are nonconstant of degree 1. If a polynomial p(x) of some degree d >1 in Q(x) has all roots in Q, then it can be written as the product of some k \leq d factors with roots in Q, and is therefore reducible in Q[x].

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As mentioned above,

f\in Q[x] is irreducible if and only if it has no root in Q

is *false* in general, as the degree 4 example I’ve given above illustrates. But

f\in Q[x] of degree 2 or degree three is irreducible if and only if it has no root in Q

is true. This is because in this case, the only way to be reducible if for it to have a linear factor, and hence, a root.

As you note, over an algebraically closed field like C, the only irreducible polynomials are the linear ones.

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Also in R[x], some polynomials like p(x) = x^3+x^2+x+1 = (x + 1)(x^2 + 1) have one real root, but two complex roots. So in this situation, if the polynomial has some of roots as real, then we cannot still say that it is reducible in R[x]?

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The polynomial you give is indeed reducible in R[x]. In fact, you can check easily that any polynomial of degree \geq 3 is reducible in R[x]. For odd degree polynomials, this is easily seen because of the existence of a real root. For even degree ones without a real root, factorize first over C, and observe that the roots occur in complex conjugate pairs, which can be combined to form quadratic factors.

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2 Comments

  1. Sandeep
    Posted January 16, 2011 at 9:13 am | Permalink | Reply

    If you take two different irreducible polynomials f and g in K[X], of the same degree d, and construct the fields K[X] / (f), and K[X] / (g), then is it correct that these will both be the “same” field? Because as vector spaces over K[X] they will both have the same dimension d. If they are the same, then does this mean that there is no unique irreducible polynomial f in K[X] which defines a field extension of K[X] of degree d?

    Thanks in advance.

    Sandeep.

  2. Posted January 17, 2011 at 10:15 am | Permalink | Reply

    For a given field extension F of K, it is true that there are many possible irreducible polynomials f such that F\simeq K[x]/(f). However, it is definitely *not* true that all irreducible polynomials of the same degree define isomorphic field extension. Consider x^2-2 and x^2-3 and ask yourself what kind of field extensions of Q we get in the two cases. It is actually a delicate question to determine when two irreducible polynomials give isomorphic field extensions. We will discuss this in class quite a bit.

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