Anonymous
Anonymous asked in Science & MathematicsMathematics · 4 years ago

(Field Theory) Which of these are false?

(1) If F ⊂ K, α ∈ K is algebraic over F, and α is a root of f(x) ∈ F[x] then the minimal polynomial of α over F divides f(x).

(2) If F ⊂ K, α ∈ K is a root of f(x) ∈ F[x] of degree n ≥ 1, then [F(α) : F] ≤ n.

(3) If Q ⊂ F ⊂ C and F/Q is finite, then there is a polynomial f(x) ∈ Q[x] such that F is contained in the splitting field of f(x) over Q.

(4) If F ⊂ C, then every ring homomorphism F → C fixes Q.

(5) If f(x) ∈ Q[x] is irreducible over Q with degree n, and roots α1,···,αn ∈ C, then every ring homomorphism ϕ : C → C restricts to an automorphism of Q(α1,···,αn). (i.e if ϕ : C → C is a ring homomorphism, then the map z → ϕ(z) defines an isomorphism Q(α1,···,αn) → Q(α1, ···,αn)).

(Q = rationals numbers, C = complex numbers)

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  • Awms A
    Lv 7
    4 years ago
    Favorite Answer

    (1) is true. Use the division algorithm to divide f(x) by the minimal polynomial.

    (2) is true. Note that F(α) is isomorphic to F[x] / ( m(x)F[x] ) where m(x) is the minimal polynomial.

    (3) is true. First write F = Q(α_1, α_2, ..., α_n) and then let f(x) be the product of the minimal polynomials of the α_i. [In fact, if memory serves, we can write F = Q(α) for a single α, but this is not obvious, to the point where I may have forgotten an assumption on that theorem]

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    Both (4) and (5) could be true or false, depending on your definition of the category of rings and ring homomorphisms. To see which definition you're using, answer these:

    (a) Is a ring required to have a unity '1' element?

    (b) Is a ring homomorphism ϕ : F --> K required to fix the unity element? i.e. Is ϕ(1_F) = 1_K ?

    (c) Is {0} a subring of Q?

    If you said

    (a) yes; (b) yes; (c) no

    then congratulations, both (4) and (5) are true as well!

    If you said

    (a) no; (b) no; (c) yes

    then congratulations to you as well, but in your case, both (4) and (5) are false!

    If you said some other combination, you need to review your definitions, as you made a mistake answering those questions.

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