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Linear and Abstract Algebra
Algebraic Extensions - Lovett, Example 7.2.7 .... ....
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[QUOTE="fresh_42, post: 5759458, member: 572553"] Well, ##m_{\alpha , \mathbb{Q}}## is irreducible. By splitting field I meant a field extension, where it is possible to factor it. But it's sufficient here to assume a factorization. A "split" into linear factors isn't needed. In any case it's important to keep in mind, that irreducibility depends on "where in". E.g. ##x^2+1## is irreducible over the reals, but as ##x^2+1=(x+i)(x-i)## it is not over ##\mathbb{C}##, i.e. it splits. Decomposition simply means a factorization, usually into linear factors, but as it's not a mathematical term, I used it as synonym for factorization (here into quadratic factors). Splitting field on the other hand is a defined term. It means the field, that provides the elements (numbers) needed to split a polynomial into linear factors, to decompose it. [/QUOTE]
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Linear and Abstract Algebra
Algebraic Extensions - Lovett, Example 7.2.7 .... ....
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