# 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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