Download Math 616 1. Corollary 29. If F is a field and p(x) ∈ F[x] is irreducible

Math 616
1. Corollary 29. If F is a field and p(x) ∈ F [x] is irreducible, then the quotient ring F [x]/(p(x)) is a
field containing (an isomorphic copy of) F and a root of F.
2. Lemma 32. Let F be a field of characteristic p > 0.
(i) For all a ∈ F , we have pa = 0.
(ii) (a + b)p = ap + bp for all a, b ∈ F .
k
k
k
(iii) (a + b)p = ap + bp for all a, b ∈ F and for all k ≥ 1.
3. If F is a field of characteristic p and if q = pk , then the function a 7→ aq is a ring homomorphism
from F to itself.
4. Theorem 33 (Galois). For every prime p and every positive integer n, there exists a field having
exactly pn elements.
5. Lemma 44. Let E/F be a field extension, let α ∈ E, and let p(x) ∈ F [x] be a monic irreducible
having α as a root.
(i) ∂(p) ≤ ∂(f ) for every f (x) ∈ F [x] having α as a root.
(ii) p(x) is the only monic polynomial in F [x] of degree ∂(p) that has α as a root.
6. Theorem 45. Let p(x) ∈ F [x] be an irreducible polynomial of degree d. Then
E = F [x]/(p(x)) is a field extension of F of degree d.
Indeed, E contains a root α of p(x), and a basis of E as a vector space over F is {1, α, α2 , . . . , αd−1 }.
7. Theorem 46. If E/F is a finite extension, then it is an algebraic extension.
8. Theorem 47. Let E/F be a field extension, and let α ∈ E be algebraic over F.
(i) There is a monic irreducible polynomial p(x) ∈ F [x] having α as a root;
∼ F (α); in fact, there is an isomorphism Φ : F [x]/(p(x)) 7→ F (α), fixing F
(ii) F [x]/(p(x)) =
pointwise, with Φ(x + (p)) = α.
(iii) p(x) is a unique monic irreducible polynomial of least degree in F [x] having α as a root.
(iv) [F (α) : F ] = ∂(p).
9. Lemma
50.P Let σ : F 7→ F 0 be an isomorphism of fields, let σ ? : F [x] −→ F 0 , defined by
P
i
ri x 7→
σ(ri )xi be the corresponding isomorphism of rings, let p(x) ∈ F [x] be irreducible,
?
and let p (x) ∈ F 0 [x].
If β is a root of p(x) and β 0 is a root of p? (x), then there is a unique isomorphism σ
b : F (β) −→ F 0 (β)
0
extending σ with σ
b(β) = β .
10. Theorem 51. Let σ : F 7→ F 0 be an isomorphism of fields, let f (x) ∈ F [x], and let f ? (x) = σ ? (f (x))
be the corresponding polynomial in F 0 (x); let E be the splitting field of f (x) over F and let E 0 be
the splitting field of f (x) over F 0 .
(i) There is an isomorphism σ
b extending σ.
(ii) If F is separable, then σ has exactly [E : F ] extensions σ
b.
11. Corollary 52. If f (x) ∈ F [x], then any two splitting fields of f (x) over F are isomorphic by an
isomorphism fixing F pointwise.
12. Corollary 52. Any two finite fields are isomorphic.