### Berkovich spaces, Problem List 1

### Let F be a field and k · k : F → R

>0### be a function.

### 1. Suppose that k0k = 0 and that for all a, b ∈ F we have

### • kabk = kakkbk,

### • k1 + ak 6 1 + kak.

### Show that k · k is a semi-norm.

### 2. Suppose that k · k is a semi-norm on F which is equivalent to the trivial norm. Show that k · k is the trivial norm.

### 3. Assume that k · k is a norm. Show that the following are equivalent.

### (a) The norm k · k is Archimedean.

### (b) For all x ∈ F

^{∗}

### there is n ∈ N such that kn · xk > 1.

### (c) There is n ∈ N such that kn · 1k > 1.

### (d) The function

### N 3 n 7→ kn · 1k ∈ R is unbounded.

### 4. Show that if k · k is a norm which is equivalent to an Archimedean norm, then k · k is Archimedean.

### 5. Suppose that k · k is an Archimedean norm on F . Show that char(F ) = 0.

### 6. Let p, q be different prime numbers. Show that | · |

p### and | · |

q### are non- equivalent non-Archimedean norms.

### 7. Suppose that k · k is a non-Archimedean non-trivial norm on Q. Show that (a) there is a prime number p such that

### pZ = {a ∈ Z | kak < 1};

### (b) the norm k · k is equivalent to the p-adic norm | · |

p### .

### 8. Suppose that k · k is an Archimedean norm on Q. Show that k · k is equivalent to the absolute value | · |

_{∞}

### .

### 9. Suppose that k · k is a non-trivial norm on F (X) which is trivial on F . Show that k · k is equivalent to | · |

_{∞}

### or there is an irreducible polynomial f ∈ F [T ] such that k · k is equivalent to | · |

_{f}

### .

### 10. Let k·k be a norm, (F, d

_{k·k}

### ) be the corresponding metric space and ( b F , b d

_{k·k}

### ) be its (metric) completion. Find a structure +, b b ·, d k · k of a normed field on F such that b

### d b

_{k·k}

### = d

k·kc