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DEFINITION 2. A field K is called a splitting field of a character 𝜒 of a group G if 𝜒∈ operatorname Char_K(G), i.e., 𝜒 is afforded by a K-representation of G. Let T be a representation of G affording the character 𝜒. It follows from Definition 2 that K is a splitting field of 𝜒 if and only if T is equivalent to 𝛥, where 𝛥 is a K-representation of G. In other words, K is a splitting field of a character 𝜒 if and only if a representation T affording 𝜒 is realized over K. Every character of G has a splitting field (for example, C is a splitting field of any character of G). If K is a splitting field of both characters 𝜒₁,𝜒₂, then K is a splitting field of 𝜒₁+𝜒₂, Therefore, in studying splitting fields, we may consider irreducible characters only. DEFINITION 3. A field K is called a splitting field of a group G if it is a splitting field for every 𝜒∈ operatorname Irr(G).
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