A bit of Galois Theory

Algebra Level 3

Let \(\mathbb Q[x]\) denote the set of polynomials with coefficients in \(\mathbb Q\).

Say \(p(x) = \sum_{k=0}^{k=n} \lambda_k x^k\), \(\lambda_n \neq 0\). We say the degree of \(p(x)\), denoted by \(deg(p)\), is \(n\).

We commonly say that a polynomial \(p(x)\) is irreducible in \(\mathbb Q[x]\), if :

\((1) \qquad p \neq 0\)

\((2) \qquad deg(p) >0 \)

\( (3) \qquad if \ \exists q,h \in \mathbb Q[x] \ such \ that \ p(x)=q(x)h(x), then \ either\)

\(\ deg(q) \ or \ deg(h) \ is \ zero\) (that is, \( \ one \ of \ \ h \ and \ g \ is \ a \ constant \))

Now, consider : \(x^4-2 \in \mathbb Q[x]\). Is this polynomial irreducible in \(\mathbb Q[x] \)? If you think it is, let \(a = -1\), else, let \(a =1 \). Same question for \(x^4+4\). If you think it is irreducible, take \(b=0\), else \(b=3\).

What is \(a+b\)?

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