myisland8132 發問於 科學及數學數學 · 1 十年前

splitting field problem

Determine the splitting field over Q of the polynomial x^4-5x^2+6 and find the degree over Q

1 個解答

  • 貓朋
    Lv 5
    1 十年前

    x^4-5x^2+6 = (x^2 -3)(x^2 -2) =(x -rt3 )(x +rt3)(x -rt2)(x +rt2)

    x^2 -3 and x^2 -2 are minimal irreducible polynomial over Q and both of them are of degree 2.

    Hence, the splitting field is Q( rt3 , rt2) and the degree over Q is 2x2=4, where rt3 and rt2 are the positive square root of 3 and 2 respectively.